Pseudospectra in Banach Jordan Algebras

1. Introduction

When an operator is not self-adjoint, its spectrum isn’t considered as the right object to study since it is very unstable under perturbation. This suggests that in many cases, the spectrum of an operator alone may not be enough to acquire a sufficient understanding of its behavior. Pseudospectra are sets in the complex plane that are designed to reveal such situations and give more information.

Let $T:\mathcal{H}⟶\mathcal{H}$ be a closed densely defined operator where $\mathcal{H}$ is a complex Hilbert space. For $ϵ>0,$ we define the $ϵ$-pseudospectrum by

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(T\right)=\mathrm{Sp}\left(T\right)\cup \left\{{z\in \mathbb{C}\setminus \mathrm{Sp}\left(T\right);\parallel (z\mathbf{1}-T)}^{-1}\parallel >{\displaystyle \frac{1}{ϵ}}\right\},\end{array}$$

(1)

where $\mathrm{Sp}\left(T\right)$ denotes the spectrum of $T.$ The main idea about these new subsets is that it is interesting to study not only the points where the resolvent of an operator is not defined (i.e., its spectrum) but also where this resolvent is large in norm. There exists an abundant literature on pseudospectra, an extensive discussion of theorems and applications together with a review of the history of pseudospectra can be found in [28].

It was shown in [15] that studying the pseudospectra of an operator is exactly studying the level lines of the norm of its resolvent. Studying such level lines gives some information about the spectral stability of the operator. Interestingly, pseudospectra can be defined in an equivalent way in terms of spectra of perturbations of the operator. For instance, we have for any matrix $\mathbf{m}\in {\mathcal{M}}_n\left(\mathbb{C}\right),$

$${\mathrm{Sp}}_{ϵ}\left(\mathbf{m}\right)=\left\{z\in \mathbb{C}:z\in \mathrm{Sp}(\mathbf{m}+\mathbf{p}),\mathrm{w}here\mathbf{p}\in {\mathcal{M}}_n\left(\mathbb{C}\right),\mathrm{w}ith\Vert \mathbf{p}\Vert \le ϵ\right\}.$$

Thus, a complex number z belongs to the $ϵ$-pseudospecrum of a matrix $\mathbf{m},$ if and only if it belongs to the spectrum of one of its perturbations $\mathbf{m}+\mathbf{p}$ with $\Vert \mathbf{p}\Vert \le ϵ.$

More generally, if T is a closed unbounded linear operator with a dense domain on a complex Hilbert space $\mathcal{H},$ Proposition 4.15 [25], states that pseudospectra can also be defined in terms of perturbation of the spectrum

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(T\right)=\bigcup _{S\in \mathcal{B}\left(\mathcal{H}\right),\parallel S\parallel <ϵ}\mathrm{Sp}(T+S),\end{array}$$

(2)

where $\mathcal{B}\left(\mathcal{H}\right)$ stands for the Banach algebra of bounded linear operators on $\mathcal{H}.$ Thus, a complex number z is in $ϵ$-pseudospectrum of $T,$ if and only if it is in the spectrum of some perturbed operator $T+S$ with $\Vert S\Vert <ϵ.$ This shows that ${\mathrm{Sp}}_{ϵ}$ is a region of instability. The importance of this formula is summarized in the following statement: Very large pseudospectra are always associated with eigenvalues, which are very instable with respect to perturbations. Obviously, this is of great importance to numerical analysts [14]: if a spectral problem is unstable enough, no numerical procedure can enable one to find the eigenvalues, whose significance therefore becomes a moot point.

When T is a closed unbounded normal linear operator with a dense domain on a complex Hilbert space $\mathcal{H},$ an exact expression for the norm of its resolvent is given in [13] as

$$\begin{array}{c}\hfill {\Vert (z\mathbf{1}-T)}^{-1}\Vert ={\displaystyle \frac{1}{d(z,\mathrm{Sp}(T\left)\right)}},\forall z\notin \mathrm{Sp}\left(T\right)\end{array}$$

(3)

where $d(z,\mathrm{Sp}(T\left)\right))$ stands for the distance between z and the spectrum of $T.$

This formula proves that the resolvent of a normal operator cannot blow up far from its spectrum. It ensures the stability of its spectrum under small perturbations because when T is selfadjoint or more generally normal, we have

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(T\right)=\left\{z\in \mathbb{C}:d(z,\mathrm{Sp}(T\left)\right)\le ϵ\right\},\end{array}$$

(4)

but in general, ${\mathrm{Sp}}_{ϵ}\left(T\right)$ is much larger than the set in the right hand side in the above equation.

Therefore, in the case of non-self-adjoint operators, we may lose the equality and have only

$$\begin{array}{c}\hfill {\parallel (z\mathbf{1}-T)}^{-1}\parallel \ge {\displaystyle \frac{1}{d(z,\mathrm{Sp}(T\left)\right)}}.\end{array}$$

(5)

The next example, taken from [17], illustrates that the spectrum of non-selfadjoint operators is not sufficient for analyzing the behaviour of the associated semigroup. A typical question that is easy in the self-adjoint case, but difficult in the non-self-adjoint case is the following: if we consider a semi-bounded self-adjoint operator A on a Hilbert space $\mathcal{H},$ we define the operator $e^{-tA}$ (i.e. $f\left(A\right)$ with $f\left(\lambda \right)=e^{-t\lambda }$ ) by the analytic functional calculus, thus we immediately obtain

$$\begin{array}{c}\hfill \parallel e^{-tA}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}=e^{-t inf \mathrm{Sp}\left(A\right)}.\end{array}$$

(6)

For example, in the case where $\mathcal{H}={\mathbb{C}}^2$ and $\mathbf{m}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ we have $\mathrm{Sp}\left(\mathbf{m}\right)=\left\{0\right\},$ but $e^{-t\mathbf{m}}=\left(\begin{array}{cc}1& -t\\ 0& 1\end{array}\right)$ does not satisfy (1.3) and (1.6). In fact, ${\parallel (\mathbf{m}-z)}^{-1}\parallel$ behaves like ${\displaystyle \frac{1}{{\left|z\right|}^2}}$ as $z⟶0.$ Hence, we need new concepts in the non-self-adjoint case.

Motivated by [8] and [12], our main aim in this paper is to extend the concepts of pseudospectrum to Banach Jordan algebras. To do so, we define $ϵ$-invertibility in Banach Jordan algebras and give the definitions of pseudospectrum of an element in a Banach Jordan algebra. The elementary properties of the pseudospectrum were studied [1] and [19]. Because of non-associativity of multiplication, the proofs in the context of Banach Jordan algebras are not just easy modifications of associative ones, but often new proofs and arguments have to be found. Besides, there are also some differences in the statements of the results in the setting of non-associative complete normed algebras compared to their counterparts established for Banach algebras.

We first establish invariance of pseudospectra in full subalgebras (Theorem 1), followed by a summary of properties of the spectrum of an element in a Banach Jordan algebra (Theorem 3). Applying the analytical functional calculus to the pseudospectrum, we obtain a pseudospectral bound on the norm of the image of an element $a\in \mathcal{J}$ under analytic function (Theorem 6) and as a corollary we characterize scalar elements in a Banach Jordan algebra. Then we consider the level sets of functions

$${\mathrm{\Psi }}_a\left(z\right)={\parallel {(a-z\mathbf{1})}^{-1}\parallel }^{-1}$$

which also determine the pseudospectrum of an element a of a Banach Jordan algebra. We observe that for real Jordan algebras, ${\mathrm{\Psi }}_a,$ is semiconvex as defined in [9].

Pseudospectral technology is well established today and highly successful as a tool for diagnosing situations where eigenvalues may be misleading. When the eigenvalues fail, however, do pseudospectra provide the missing information? Are they a qualitatively helpful tool, or merely a warning device? Many results pertaining to this situation are collected in [28], $\S \S 14-19,$ but a general answer is not clear. For example, suppose A and B are two matrices with simple eigenvalues whose $ϵ$-pseudospectra are identical for all $ϵ.$ Must $\parallel p\left(A\right)\parallel = \parallel p\left(B\right)\parallel$ for all polynomials $p?$ Without the hypothesis of simple eigenvalues, the answer is no; see [28], $\S 47.$ In Proposition 4, we consider the situation of two elements a and b in a Banach Jordan algebra $\mathfrak{J}$ such that ${\mathrm{\Psi }}_a\left(z\right)={\mathrm{\Psi }}_b\left(z\right)$.

A new concept of instability index of an isolated spectral value is defined and some of its general properties are outlined in the context of Banach Jordan algebras. Then, we turn to the study of linear maps preserving pseudospectra in Banach Jordan algebras, extending similar results as in [19].

In the last part of the paper, we examine whether there exists a decomposition of the pseudospectrum ${\mathrm{Sp}}_{ϵ}(\mathfrak{J},x)$ of an element x in a Banach Jordan algebra $\mathfrak{J}$ in terms of pseudospectra ${\mathrm{Sp}}_{ϵ}(U_p\mathfrak{J},U_{p}x)$ and ${\mathrm{Sp}}_{ϵ}(U_q\mathfrak{J},U_{q}x),$ where p and q are orthogonal projections in $\mathfrak{J}$ (Theorem 12). Finally, we partially extend to JB-algebras, an important theorem previously proved by Roch-Silbermann in the context of $C^{*}$-algebras, stating that the pseudospectrum is a region of spectral instability.

2. Preliminaries

2.1. Jordan Algebras

A Jordan algebra $\mathfrak{J}$ is a non-associative algebra whose product · satisfies $a·b=b·a$ and $(a·b)·a^2=a·(b·a^2)$ for all $a,b\in \mathfrak{J},$ where $a^2=a·a.$ For $x\in \mathfrak{J},$ the operator of left multiplication $L_x$ is defined by $L_{x}a:=x·a$ for all $a\in \mathfrak{J}.$ The operator of right multiplication obviously coincides with $L_x.$ As usual, $U_{a,b}$ denotes the operator on J defined by

$$U_{a,b}x=\{a,x,b\}=(L_a{L}_b+L_b{L}_a-L_{a·b})x$$

and we set $U_a=U_{a,a}$ for all $a,b,x\in \mathfrak{J}.$ The identity element (or the unit) in $\mathfrak{J}$ is an element $\mathbf{1}$ such that $a·\mathbf{1}=a$ for each $a\in \mathfrak{J}.$ Let ${\mathfrak{J}}^1$ denote $\mathfrak{J}$ if $\mathfrak{J}$ is unital, and the Jordan algebra $\mathfrak{J}\oplus \mathbb{C}$ obtained from J by adjoining the unit $\mathbf{1}$ otherwise. An element a in a unital Jordan algebra $\mathfrak{J}$ is invertible if there is $b\in \mathfrak{J}$ such that $a·b=\mathbf{1}$ and $a^2·b=a.$ The inverse is unique and denoted by $a^{-1}.$ The set of all invertible elements of $\mathfrak{J}$ is identified by $\mathrm{Inv}\left(\mathfrak{J}\right).$ An element $a\in \mathfrak{J}$ is invertible in $\mathfrak{J}$ if and only if $U_a$ is invertible, and in this case $U_a^{-1}=U_{a^{-1}}.$ The fundamental identity

$$U_{U_{a}b}=U_a{U}_b{U}_a$$

for all $a,b\in \mathfrak{J}$ shows that $U_{a}b$ is invertible if and only if a and b are invertible. Taking into account that $a^{-1}=U_{a^{-1}}a=U_a^{-1}a$ for each $a\in \mathrm{Inv}\left(\mathfrak{J}\right),$ we see from the above that

$$\begin{array}{cc}\hfill {\left(U_{a}b\right)}^{-1}& =U_{U_{a}b}^{-1}{U}_{a}b\hfill \\ \hfill & =U_{a^{-1}}{U}_{b^{-1}}{U}_{a^{-1}}{U}_{a}b\hfill \\ \hfill & =U_{a^{-1}}{b}^{-1}\hfill \end{array}$$

(7)

for all $a,b\in \mathrm{Inv}\left(\mathfrak{J}\right).$ It is easy to check that

$$U_{a-\lambda }=U_a-2\lambda L_a+{\lambda }^2$$

for all $a\in \mathfrak{J}$ and $\lambda \in \mathbb{C}.$ In particular, $U_1=U_{-1}$ is the identity operator on $\mathfrak{J}.$

For $x\in \mathfrak{J},$ the spectrum of x (denoted by ${\mathrm{Sp}}_{\mathfrak{J}}\left(x\right)$ or simply $\mathrm{Sp}\left(x\right)$) is the set of complex numbers $\lambda$ for which $\lambda ·\mathbf{1}-\mathbf{x}$ is not invertible in ${\mathfrak{J}}^1.$ We have by definition, ${\mathrm{Sp}}_{\mathfrak{J}}\left(x\right)={\mathrm{Sp}}_{{\mathfrak{J}}^1}\left(x\right).$

A subalgebra $\mathfrak{L}$ of $\mathfrak{J}$ is called strongly associative if $[L_a,L_b]=0$ for any $a,b\in \mathfrak{L}$, spectral if ${\mathrm{Sp}}_{\mathfrak{L}}\left(a\right)\cup \left\{0\right\}={\mathrm{Sp}}_{\mathfrak{J}}\left(a\right)\cup \left\{0\right\}$ for each $a\in \mathfrak{L},$ and full (or inverse closed) if $\mathfrak{L}$ contains the inverse of each element of $\mathfrak{L}$ that is invertible in $\mathfrak{J}.$Note that each strongly associative subalgebra is associative. For any $a\in \mathfrak{J}$ there is a maximal strongly associative subalgebra containing a which is inverse closed whenever $\mathfrak{J}$ is unital see [16]; this subalgebra is spectral.

The following well-known proposition illustrates the importance of strongly associative algebras in studying problems related to the spectrum in Jordan algebras.

Proposition 1.

Let $\mathfrak{J}$ be a unital Jordan algebra, and let $x,a\in \mathfrak{J}$ be invertible. Then

(i) 

$\mathrm{Sp}\left(a^{-1}\right)=\{{\lambda }^{-1}:\lambda \in \mathrm{Sp}\left(a\right)\}.$

(ii) 

$\mathrm{Sp}\left(U_a{x}^2\right)=\mathrm{Sp}\left(U_x{a}^2\right).$

Proof. (i) Let $\mu \in \mathbb{C}$ be non-zero. As $a-\mu =U_a(a^{-1}-\mu a^{-2}),$ we see that $a-\mu$ and $a^{-1}-\mu a^{-1}$ are simultaneously invertible or not. Let $\mathfrak{A}$ be a maximal strongly associative subalgebra of $\mathfrak{J}$ containing $a.$ Then $a^{-1}-\mu a^{-2}$ is invertible in $\mathfrak{J}$ if and only if it is so in $\mathfrak{A}.$ As $\mathfrak{A}$ is an associative algebra, $a^{-1}-\mu a^{-2}$ is invertible in $\mathfrak{A}$ if and only if $1-\mu a^{-1}$ is invertible in $\mathfrak{A}.$ Since $\mathfrak{A}$ is an inverse closed subalgebra of $\mathfrak{J},$ it follows that $1-\mu a^{-1}$ is invertible in $\mathfrak{A}$ if and only if it is so in $\mathfrak{J}.$ Hence,

$$\mathrm{Sp}\left(a^{-1}\right)=\{{\lambda }^{-1}:\lambda \in \mathrm{Sp}\left(a\right)\}.$$

(ii) It is clear that $U_x{a}^2$ and $U_a{x}^2$ are invertible. Let $\lambda \in \mathbb{C}$ be non-zero. Then

$$U_x{a}^2-\lambda =U_x(a^2-\lambda x^{-2})=U_x{U}_a(1-U_{a^{-1}}{x}^{-2})$$

implies that $U_x{a}^2-\lambda$ is invertible if and only if $1-\lambda U_{a^{-1}}{x}^{-2}$ is invertible. As $U_{a^{-1}}{x}^{-2}$ is the inverse of $U_a{x}^2,$ we have that $\mathrm{Sp}\left(U_a{x}^2\right)=\mathrm{Sp}\left(U_x{a}^2\right).$ □

2.2. Complete Normed Jordan Algebras

A Jordan algebra is called normed if it is a normed space with norm $\parallel ·\parallel$ and $\parallel x·y\parallel \le \parallel x\parallel \parallel y\parallel$ for all $x,y\in \mathfrak{J}.$ If $\mathfrak{J}$ is complete under the norm, then $\mathfrak{J}$ is called a Banach (or complete normed) Jordan algebra. Let $\mathfrak{J}$ be a unital normed Jordan algebra and ${\mathcal{A}}^{ic}\left(x\right)$ be a closed full subalgebra of $\mathfrak{J}$ generated by $x\in \mathfrak{J}.$ It follows from [16] that ${\mathcal{A}}^{ic}\left(x\right)$ is associative and is the closure of the algebra of rational functions $p\left(x\right)·q{\left(x\right)}^{-1},$ where $p,q$ are polynomials and $q\left(x\right)$ is invertible. By [16], in a Banach Jordan algebra $\mathfrak{J},$ we know that ${\mathrm{Sp}}_{\mathfrak{J}}\left(x\right)={\mathrm{Sp}}_{{\mathcal{A}}^{ic}\left(x\right)}\left(x\right)$ is a compact non-empty set in $\mathbb{C}$ and the function $\lambda \mapsto {(\lambda -x)}^{-1}$ is analytic on $\mathbb{C}\setminus \mathrm{Sp}\left(x\right).$ By analytic functional calculus see [16], for any $\mathbb{C}$-valued function f analytic on some neighborhood V of $\mathrm{Sp}\left(x\right)$ there is an element $f\left(x\right)\in {\mathcal{A}}^{ic}\left(x\right)$ defined by

$$f\left(x\right)={\left(2\pi i\right)}^{-1}{\int }_{\mathrm{\Gamma }}f\left(\xi \right){(\xi \mathbf{1}-x)}^{-1}d\xi$$

where $\mathrm{\Gamma }$ is a suitable contour in V surrounding $\mathrm{Sp}\left(x\right)$ and $f\left(x\right)$ does not depend on the choice of V and $\mathrm{\Gamma }.$ The map $f\mapsto f\left(x\right)$ is a homomorphism from the algebra $\mathcal{O}\left(V\right)$ of functions analytic on V into $J;$ this homomorphism is continuous with respect to convergence of functions on compact subsets of V and satisfies for every function f analytic in some open neighborhood of $\mathrm{Sp}\left(x\right),$

$$f\left(\mathrm{Sp}\right(x\left)\right)=\mathrm{Sp}\left(f\right(x\left)\right)$$

If $\mathfrak{J}$ is a Banach Jordan algebra, then the map $x\mapsto \mathrm{Sp}\left(x\right)$ is upper semicontinuous see [5] or [16]. The following results were given in [16] as Lemma 4.1.4 and Proposition 4.1.92 respectively.

3. $ϵ$-Invertibility in Banach Jordan Algebras

3.1. Add Text

Let $\mathfrak{J}$ be a unital complete normed Jordan algebra over $\mathbb{C}.$ We recall that the spectrum of an element $a\in \mathfrak{J}$ denoted by $\mathrm{Sp}(a,\mathfrak{J})$ or simply $\mathrm{Sp}\left(a\right)$ is defined as

$$\begin{array}{c}\hfill \mathrm{Sp}\left(a\right):=\{\lambda \in \mathbb{C}:\lambda \mathbf{1}-a\notin \mathrm{Inv}(\mathfrak{J}\left)\right\},\end{array}$$

where $\mathrm{Inv}\left(\mathfrak{J}\right)$ is the set of all invertible elements of $\mathfrak{J}.$ Let $\widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$ be the extended complex plane. As in [21] one can define the function $\mathbf{nr}:\mathfrak{J}\times \mathbb{C}\to [0,\infty ]$ (the norm of the resolvent) by the formula

$$\mathbf{nr}(a,\lambda )=\left\{\begin{array}{ccc}{\parallel (\lambda \mathbf{1}-a)}^{-1}\parallel \hfill & \mathrm{if}& \hfill \lambda \in \mathbb{C}\setminus \mathrm{Sp}\left(a\right)\\ +\infty \hfill & \mathrm{if}& \hfill \lambda \in \mathrm{Sp}\left(a\right)\\ 0\hfill & \mathrm{if}& \hfill \lambda =\infty \end{array}\right.$$

It is easy to see that the function $\mathbf{nr}(a,·)$ is continuous on $\widehat{\mathbb{C}}.$ In particular,

$$|\mathbf{nr}(a,\mu )-\mathbf{nr}(a,\lambda )|<{\displaystyle \frac{{|\mu -\lambda |\mathbf{nr}(a,\lambda )}^2}{1-|\mu -\lambda |\mathbf{nr}(a,\lambda )}}.$$

Definition 1

($ϵ$-pseudospectrum). If $a\in \mathfrak{J}$ and $ϵ>0$ then the ϵ-pseudospectrum of $a\in \mathfrak{J}$ is defined to be the equality ${\mathrm{Sp}}_{ϵ}\left(a\right)=\left\{\lambda \in \mathbb{C}\mid \mathbf{nr}(a,\lambda )\ge {\displaystyle \frac{1}{ϵ}}\right\}.$

Unlike the spectrum, which is independent of the norm, the pseudospectrum depends on the norm. We will prove that ${\mathrm{Sp}}_{ϵ}\left(a\right)$ is a compact subset of $\mathbb{C}$ and contains the spectrum of $a.$

Proposition 2.

Let $\mathfrak{J}$ be a normed unital Jordan algebra and let a be in $\mathfrak{J}.$ Then the closed $\mathfrak{J}$-full subalgebra of $\mathfrak{J}$ generated by a is associative and commutative.

In associative Banach algebras, inverse closed subalgebras parallel the notion of full subalgebras defined for Jordan algebras. Indeed, if $\mathfrak{A}$ is a complex unital Banach algebra and $\mathfrak{B}$ is a unital subalgebra of $\mathfrak{A},$ we say that $\mathfrak{B}$ is inverse closed in $\mathfrak{A}$ if every element of $\mathfrak{B},$ which is invertible in $\mathfrak{A},$ is also invertible in $\mathfrak{B},$ that is $\mathfrak{B}\cap \mathrm{Inv}\left(\mathfrak{A}\right)\subseteq \mathrm{Inv}\left(\mathfrak{B}\right).$

A subalgebra B of a Banach Jordan algebra $\mathfrak{J}$ is a full subalgebra if the unit of $\mathfrak{J}$ belongs to B and $\mathrm{Inv}\left(B\right)=B\cap \mathrm{Inv}\left(\mathfrak{J}\right).$

Theorem 1.

Let $\mathfrak{J}$ be a complex unital Banach Jordan algebra with unit 1, and $\mathfrak{L}$ a full subalgebra of $\mathfrak{J}.$ Then, for all $l\in \mathfrak{L},\mathrm{Sp}(l,\mathfrak{L})=\mathrm{Sp}(l,\mathfrak{J}).$ Also, for every $ϵ>0,{\mathrm{Sp}}_{ϵ}(l,\mathfrak{L})={\mathrm{Sp}}_{ϵ}(l,\mathfrak{J}).$

Proof. 

Obviously, for each $l\in \mathfrak{L},\mathrm{Sp}(l,\mathfrak{J})\subseteq \mathrm{Sp}(l,\mathfrak{J}).$ Next, let $l\in \mathfrak{L},\lambda \in \mathbb{C}$ and $\lambda \notin \mathrm{Sp}(l,J).$ Then $\lambda ·\mathbf{1}-l$ is invertible in $\mathfrak{J}$ and hence also in $\mathfrak{L}.$ Thus $\lambda \notin \mathrm{Sp}(l,\mathfrak{L}).$ Hence $\mathrm{Sp}(l,\mathfrak{L})\subseteq \mathrm{Sp}(l,\mathfrak{J}).$ This shows that $\mathrm{Sp}(l,\mathfrak{J})=\mathrm{Sp}(l,\mathfrak{L}).$

Since $\mathfrak{L}$ inherits the norm on $\mathfrak{J}$, the norms on $\mathfrak{L}$ and $\mathfrak{J}$ are the same, it follows from the definition of pseudospectrum, that ${\mathrm{Sp}}_{ϵ}(l,\mathfrak{J})={\mathrm{Sp}}_{ϵ}(l,\mathfrak{L})$ for all $ϵ>0.$ □

3.2. Resolvent and Subharmonic Functions

Recall that a continuous function $\rho :D⟶\mathbb{R}$ on a domain $D\subset \mathbb{C}$ is subharmonic if, for all $z_0\in D$ and $r>0,$ such that the closed disk $|z-z_0| \le r$ is contained in $D,$ we have

$$\rho \left(z\right)\le {\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\rho (z_0+r{e}^{it})dt.$$

The following results are well known and could be found in either [5] or [16]. Let D be an open set in the complex plane, let E be a complex Banach space and let f be a holomorphic function on D with values in $E,$ i.e. a mapping $f:D⟶E$ such that for every continuous linear functional on $E,$ the complex valued function $\mu \circ f:D⟶\mathbb{C}$ is holomorphic.

Proposition 3

([16]). Let $D\subseteq \mathbb{C}$ be an open set, let E be a Banach space, and let $f:D⟶E$ be holomorphic. Then $\parallel f\parallel$ is subharmonic in $D.$

Theorem 2.

Let a be an element of a Banach Jordan algebra $\mathfrak{J}.$ Then the function

$${\mathrm{\Psi }\left(z\right)=\parallel (a-z\mathbf{1})}^{-1}\parallel$$

is subharmonic on $\mathbb{C}\setminus \mathrm{Sp}\left(a\right).$

Proof. 

Apply Proposition 3 to the holomorphic function $z\mapsto {(a-z)}^{-1}.$ □

4. Properties of the Pseudospectrum

It is known that ${\mathrm{Sp}}_{ϵ}\left(a\right)$ is much larger than $\mathrm{Sp}\left(a\right),$ then small perturbations can move spectral values very far. So, it is important to know whether the pseudospectra are sensitive to small perturbations. If they were, they would be of little value. Fortunately, this is not the case, as shown in the following theorem. Given an element a in a Banach Jordan algebra $\mathfrak{J},$ the pseudospectra ${\mathrm{Sp}}_{ϵ}\left(a\right)$ are nested compact subsets in the complex plane whose intersection is $\mathrm{Sp}\left(a\right).$ In what follows, $\mathbb{D}(\alpha ,r)$ stands for the disk centered at $\alpha$ and radius r in the complex plane.

Theorem 3.

Let $\mathfrak{J}$ be a complex unital Banach Jordan algebra and $a\in \mathfrak{J}.$ Then,

(i) 

$$\mathrm{Sp}\left(a\right)={\cap }_{ϵ>0}{\mathrm{Sp}}_{ϵ}\left(a\right).$$

(ii) 

$${\mathrm{Sp}}_{ϵ}(a+\lambda )=\lambda +{\mathrm{Sp}}_{ϵ}\left(a\right)$$

where $\lambda \in \mathbb{C}.$

(iii) 

$${\mathrm{Sp}}_{{ϵ}_1}\left(a\right)\subseteq {\mathrm{Sp}}_{{ϵ}_2}\left(a\right)$$

with $0<{ϵ}_1<{ϵ}_2.$

(iv) 

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(\lambda a\right)=\lambda {\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{\left|\lambda \right|}}\left(a\right)}(\lambda \in \mathbb{C}\setminus \left\{0\right\})\end{array}$$

(8)

and

$$\begin{array}{c}\hfill \alpha {\mathrm{Sp}}_{ϵ}\left(a\right)={\mathrm{Sp}}_{\left|\alpha \right|ϵ}\left(\alpha a\right),ϵ>0,\alpha \ne 0.\end{array}$$

(9)

(v) 

$${\mathrm{Sp}}_{ϵ}\left(a\right)\subseteq \mathbb{D}(0;3\parallel a\parallel +ϵ).$$

(vi) 

${\mathrm{Sp}}_{ϵ}\left(a\right)$ is a non-empty compact subset of $\mathbb{C}.$

Proof. 

(i)

Let $\lambda \in \mathrm{Sp}\left(a\right).$ Then $a-\lambda ·\mathbf{1}$ is not invertible, hence by the convention in Definition 1, ${\parallel (\lambda ·\mathbf{1}-a)}^{-1}\parallel =\infty ,$ i.e. ${\parallel (\lambda ·\mathbf{1}-a)}^{-1}\parallel >{\displaystyle \frac{1}{ϵ}}$ for all $ϵ>0.$ So, ${\displaystyle \lambda \in {\cap }_{ϵ>0}{\mathrm{Sp}}_{ϵ}\left(a\right).}$ On the other hand, if $\lambda \notin \mathrm{Sp}\left(a\right),$ then $\lambda ·\mathbf{1}-a$ is invertible. Hence there exists ${ϵ}_0>0,$ such that ${\parallel (\lambda ·\mathbf{1}-a)}^{-1}\parallel <{\displaystyle \frac{1}{{ϵ}_0}}.$ Then, $\lambda \notin {\mathrm{Sp}}_{{ϵ}_0}\left(a\right).$

(ii)

It is obvious that if $z\in \lambda +{\mathrm{Sp}}_{ϵ}\left(a\right)$ then $z\in {\mathrm{Sp}}_{ϵ}(\lambda ·\mathbf{1}+a).$ Let $z\in {\mathrm{Sp}}_{ϵ}(a+\lambda ·\mathbf{1}).$ Then

$${\parallel (z·\mathbf{1}-(a+\lambda ·\mathbf{1}))}^{-1}{\parallel = \parallel ((z-\lambda )·\mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$$

Therefore, $z-\lambda \in {\mathrm{Sp}}_{ϵ}\left(a\right),$ that is $z\in \lambda +{\mathrm{Sp}}_{ϵ}\left(a\right).$

(iii)

Let $0<{ϵ}_1<{ϵ}_2,$ and suppose $\lambda \in {\mathrm{Sp}}_{{ϵ}_1}\left(a\right).$ If $\lambda ·\mathbf{1}-a$ is not invertible, then $\lambda \in \mathrm{Sp}\left(a\right)\subseteq {\mathrm{Sp}}_{{ϵ}_2}\left(a\right).$ Otherwise, $\lambda ·\mathbf{1}-a$ is invertible and

$${\parallel (\lambda ·\mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{{ϵ}_1}}>{\displaystyle \frac{1}{{ϵ}_2}}.$$

Accordingly, $\lambda \in {\mathrm{Sp}}_{{ϵ}_2}\left(a\right).$

(iv)

Let $z\in {\mathrm{Sp}}_{ϵ}\left(\lambda a\right).$ Then,

$${\parallel (z·\mathbf{1}-\lambda ·a)}^{-1}\parallel =\parallel {({\displaystyle \frac{z}{\lambda }}·\mathbf{1}-a)}^{-1}{\parallel \left|\lambda \right|}^{-1}>{\displaystyle \frac{1}{ϵ}}.$$

So

$${\left|\lambda \right|\parallel (z·\mathbf{1}-\lambda ·a)}^{-1}\parallel \ge {\displaystyle \frac{\left|\lambda \right|}{ϵ}}.$$

Hence,

$${\displaystyle \frac{z}{\lambda }}\in {\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{\left|\lambda \right|}}}\left(a\right),$$

that is,

$$z\in \lambda {\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{\left|\lambda \right|}}}\left(a\right).$$

The second equality follows by taking $\alpha ={\displaystyle \frac{1}{\lambda }}$ and $b={\displaystyle \frac{1}{\alpha }}a$ in (4.2).

(v)

Suppose $\left|z\right|>3\parallel a\parallel +ϵ.$ In particular, $z\notin \mathbb{D}(0;\parallel a\parallel +ϵ).$ Then, $\left|z\right|>\parallel a\parallel$ and $z\mathbf{1}-a$ is invertible. It follows, from Corollary 4.1.5 in [16], that

$${\parallel (z·\mathbf{1}-a)}^{-1}\parallel \le {\displaystyle \frac{1}{\left|z\right|-3\parallel a\parallel }}<{\displaystyle \frac{1}{ϵ}}.$$

Hence

$$z\notin {\mathrm{Sp}}_{ϵ}\left(a\right).$$

So

$${\mathrm{Sp}}_{ϵ}\left(a\right)\subseteq \mathbb{D}(0;3\parallel a\parallel +ϵ).$$

(vi)

By Part (v), ${\mathrm{Sp}}_{ϵ}\left(a\right)$ is bounded. It is closed by continuity of the norm and resolvent function $R_a\left(z\right)={(z·\mathbf{1}-a)}^{-1}.$ Hence, it is compact. Since ${\mathrm{Sp}}_{ϵ}\left(a\right)$ contains $\mathrm{Sp}\left(a\right),$ it is obviously nonempty.

□

4.1. Add Text

It is shown in Theorem 5.1 and Theorem 5.2 in [19] for Banach algebras, that the pseudospectrum has no isolated points and it has a finite number of components. Similar results hold in Banach Jordan algebras and their proofs are the same because we mainly work in the local strongly associative algebra generated by 1 and $a.$

Theorem 4.

Let $\mathcal{J}$ be a Jordan Banach algebra, $a\in \mathcal{J}$ and $ϵ>0.$ Then, the pseudospectrum has no isolated points.

Theorem 5.

Let $\mathcal{J}$ be a Jordan Banach algebra, $a\in \mathcal{J}$ and $ϵ>0.$ Then, the pseudospectrum ${\mathrm{Sp}}_{ϵ}\left(a\right)$ of a has a finite number of components and each component of ${\mathrm{Sp}}_{ϵ}\left(a\right)$ contains an element of $\mathrm{Sp}\left(a\right).$

4.2. Level Sets of Functions

Similarly to [22], we define for an element a in a real Banach Jordan algebra $A,$ the level sets of functions

$${\mathrm{\Psi }}_a\left(z\right)=\left\{\begin{array}{ccc}{\Vert (a-z\mathbf{1})}^{-1}{\Vert }^{-1}\hfill & \mathrm{if}& \hfill z\notin \mathrm{Sp}\left(a\right)\\ 0\hfill & \mathrm{if}& \hfill z\in \mathrm{Sp}\left(a\right)\end{array}\right.$$

This function is closely related to the previously well-defined $ϵ$-pseudospectrum of $a,$ that is also equally expressed as

$${\mathrm{Sp}}_{ϵ}\left(a\right)=\left\{z\in \mathbb{C}:{\mathrm{\Psi }}_a\left(z\right)\le ϵ\right\}.$$

It is shown in [9] that this function is locally semiconvex. We can follow exactly the same proof as in [9] to extend this result to real Jordan algebras in a straightforward way.

4.3. Pseudospectral Bounds on $\parallel f\left(a\right)\parallel$

It is known that if $\mathbf{m}\in {\mathcal{M}}_n\left(\mathbb{C}\right)$ and f is analytic on some open neighborhood of ${\mathrm{Sp}}_{ϵ}\left(\mathbf{m}\right)$ for some $ϵ>0,$ then

$$\parallel f\left(\mathbf{m}\right)\parallel \le {\displaystyle \frac{L_{ϵ}}{2\pi ϵ}}\underset{z\in {\mathrm{Sp}}_{ϵ}\left(\mathbf{m}\right)}{sup}\left|f\left(z\right)\right|,$$

where $L_{ϵ}$ denotes the contour of the boundary of ${\mathrm{Sp}}_{ϵ}\left(\mathbf{m}\right)$ see [27].

The following theorem extends this result from matrices to elements of Banach Jordan algebras.

Theorem 6.

Let $\mathfrak{J}$ be a Banach Jordan algebra, $a\in \mathfrak{J}$ and f be analytic on an open neighbourhood $\mathrm{\Omega }\subset \mathbb{C}$ of ${\mathrm{Sp}}_{ϵ}\left(a\right).$ Then

$$f\left(a\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathrm{\Gamma }}{(z\mathbf{1}-a)}^{-1}f\left(z\right)dz$$

where Γ be a contour that surrounds ${\mathrm{Sp}}_{ϵ}\left(a\right)$ in Ω and

$$\Vert f\left(a\right)\Vert \le {\displaystyle \frac{Ml}{2\pi ϵ}}$$

where $l=$ length of Γ and $M=sup\left\{\right|f\left(z\right):z\in \mathrm{\Gamma }\}.$

Proof. 

By the analytical functional calculus Theorem,

$$f\left(a\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathrm{\Gamma }}{(z\mathbf{1}-a)}^{-1}f\left(z\right)dz$$

is well defined. Hence,

$$\Vert f\left(a\right)\Vert \le {\displaystyle \frac{1}{2\pi }}{\int }_{\mathrm{\Gamma }}{\Vert (z-a)}^{-1}\Vert \Vert f\left(z\right)\Vert \left|dz\right|\le {\displaystyle \frac{Ml}{2\pi ϵ}}$$

since $\mathrm{\Gamma }$ lies outside the interior of ${\mathrm{Sp}}_{ϵ}\left(a\right).$

□

The next corollary gives an equivalent condition in terms of the $ϵ$-pseudospectrum for an element of a Banach Jordan algebra to be a scalar,that is a multiple of the identity element.

Corollary 1.

Let $\mathfrak{J}$ be a Banach Jordan algebra and $a\in \mathfrak{J}$ and $\mu \in \mathbb{C}.$ Then

$$a=\mu \iff {\mathrm{Sp}}_{ϵ}\left(a\right)=\mathbb{D}(\mu ,ϵ)\forall ϵ>0.$$

Proof. 

If $a=\mu ,$ it is obvious to see that ${\mathrm{Sp}}_{ϵ}\left(a\right)=\mathbb{D}(\mu ,ϵ),\forall ϵ>0.$

For the converse, by part (ii) of Theorem 3, we may assume that $\mu =0.$ Let $f\left(z\right)=z$ and $\mathrm{\Gamma }=\{z\in \mathbb{C}:|z|=ϵ\}.$ Then, with the notations of Theorem 6, we have $M=ϵ$ and $l=2\pi ϵ.$ We conclude that $\Vert a\Vert \le ϵ.$ Because this is true for all $ϵ>0,$ we get $a=0=\mu .$ □

Using the analytic functional calculus and the level sets functions, we extend the following result proved in [24] for associative Banach algebras to complete normed non-associative algebras. This solves the question on the relation between two elements a and b such that ${\mathrm{\Psi }}_a\left(z\right)={\mathrm{\Psi }}_b\left(z\right)$ for $z\in \mathbb{C}.$

Proposition 4.

Let $(\mathfrak{J},\parallel ·\parallel )$ be a unital complex Banach Jordan algebra and $a,b\in \mathfrak{J}$ such that

$${\parallel (a-z\mathbf{1})}^{-1}{\parallel }^{-1}={\parallel {(b-z\mathbf{1})}^{-1}\parallel }^{-1}(z\in \mathbb{C}).$$

Then

$$\parallel a\parallel \le e\parallel b\parallel .$$

Proof. 

Let $r>\parallel b\parallel$ and $n>{\displaystyle \frac{\parallel a\parallel }{r}}.$ Integrating the Taylor series associated to the function $\zeta \mapsto {\left(1-{\displaystyle \frac{a}{n\zeta }}\right)}^{-n},$ we obtain the integral representation

$$a={\displaystyle \frac{1}{2\pi i}}{\int }_{\left|\zeta \right|=r}{\left(1-{\displaystyle \frac{a}{n\zeta }}\right)}^{-n}d\zeta .$$

Consequently,

$$\begin{array}{cc}\hfill \parallel a\parallel & \le r\underset{\left|\zeta \right|=r}{sup}{\parallel {\left(1-{\displaystyle \frac{a}{n\zeta }}\right)}^{-1}\parallel }^n\hfill \\ \hfill & =r\underset{\left|\zeta \right|=r}{sup}{\parallel {\left(1-{\displaystyle \frac{b}{n\zeta }}\right)}^{-1}\parallel }^n\hfill \\ \hfill & \le r{\left(1-{\displaystyle \frac{\parallel b\parallel }{nr}}\right)}^{-n}.\hfill \end{array}$$

(10)

Letting $r⟶\parallel b\parallel$ and $n⟶\infty ,$ it follows that $\parallel a\parallel \le e\parallel b\parallel .$

□

4.4. Pseudospectrum and Instability Index

Every isolated spectral value of an element $a\in \mathfrak{J},$ is associated with a quantity called index of instability which measures the instability of this spectral value under small perturbations of $a.$ If $\lambda$ is an isolated point of the spectrum of an element $a\in \mathfrak{J},$ the spectral projection associated with $\lambda$ is defined (see [5] or [7]) by ${\displaystyle \mathbf{p}=\frac{1}{2\pi i}{\int }_{\gamma }{(z\mathbf{1}-a)}^{-1}dz,}$ where $\gamma$ is any sufficiently small closed contour winding around $\lambda ,$ separating $\lambda$ from the rest of the remaining spectrum of $a.$ In fact, $\mathbf{p}$ does not depend on the contour $\gamma ,$ as long as $\gamma$ separates $\lambda$ from the rest of the spectrum. Thus, we can suppose that $\gamma$ is a small circle with centre at $\lambda .$ The assumption that this projection has rank 1 is stronger than the assumption that $\lambda$ is an eigenvalue of multiplicity 1 (see [7] for the definition of spectral rank and spectral multiplicity). We extend the definition of the index of instability of Aslanyan and Davies in [4] from Schrödinger operators to the general context of elements in a Banach Jordan algebra.

Definition 2.

Let a be an element in a Banach Jordan algebra $\mathfrak{J}$ and λ an isolated spectral value of $a.$ The index of instability associated to λ is defined as

$$\kappa \left(\lambda \right)=\parallel \mathbf{p}\parallel ,$$

where $\mathbf{p}$ is the spectral projection associated to $\lambda .$

The instability index is also related to the notion of pseudospectrum, which is a geometric way of looking at resolvent norm properties. Namely, if $ϵ>0$ we recall that

$${\mathrm{Sp}}_{ϵ}\left(a\right)=\left\{{z\in \mathbb{C}:\parallel (z\mathbf{1}-a)}^{-1}\parallel \ge {ϵ}^{-1}\right\}.$$

The sizes of these sets, which all contain the spectrum of $a,$ measure the spectral instability of $a.$ One always has

$$\{z:\mathrm{dist}\{z,\mathrm{Sp}\left(a\right)\}\}<ϵ\}\subseteq {\mathrm{Sp}}_{ϵ}\left(a\right),$$

but the RHS is often much larger than the LHS. The following theorem states that if $\kappa \left(\lambda \right)$ is large, then the component of ${\mathrm{Sp}}_{ϵ}\left(a\right)$ containing $\lambda$ is large in a related sense. It also establishes a connection between pseudospectra and the norms of the spectral projections.

Theorem 7.

Suppose that the spectral projection $\mathbf{p}$ associated with the isolated specral value λ of a has rank 1. Let γ be a closed contour surrounding the connected component of ${\mathrm{Sp}}_{ϵ}\left(a\right)$ which contains λ but does not intersect ${\mathrm{Sp}}_{ϵ}\left(a\right).$ Then,

$$\left|\gamma \right|\ge 2\pi ϵ\kappa \left(\lambda \right)$$

where $\left|\gamma \right|$ is the length of $\gamma .$

Proof. 

We have from the definition

$$\kappa \left(\lambda \right)=\parallel \mathbf{p}\parallel =\parallel {\displaystyle \frac{1}{2\pi i}}{\int }_{\gamma }{(z\mathbf{1}-a)}^{-1}dz\parallel \le {\displaystyle \frac{1}{2\pi ϵ}}{\int }_{\gamma }\left|dz\right|={\displaystyle \frac{\left|\gamma \right|}{2\pi ϵ}}.$$

□

Remark 1.

It follows from the preceding theorem that one good reason to study the index of instability associated to isolated spectral values of an element a in a Banach Jordan algebra $\mathfrak{J}$ is linked to the size of the pseudospectrum surrounding the corresponding spectral value. Indeed, let ${\mathrm{Sp}}_{ϵ}^{\left(\lambda \right)}$ be the connected component of ${\mathrm{Sp}}_{ϵ}\left(a\right)$ containing the isolated spectral value $\lambda ,$ and suppose that this component is bounded and does not contain any other point of the spectrum of $a.$ By choosing any contour γ enclosing ${\mathrm{Sp}}_{ϵ}^{\left(\lambda \right)}$ without surronding any other point of ${\mathrm{Sp}}_{ϵ}\setminus {\mathrm{Sp}}_{ϵ}^{\left(\lambda \right)},$ and not intersecting ${\mathrm{Sp}}_{ϵ}\left(a\right),$ it follows from Theorem 8 that

$$\left|\gamma \right|\ge 2\pi ϵ\kappa \left(\lambda \right).$$

5. Linear Maps Preserving Pseudospectrum in Banach Jordan Algebras

Some results obtained in [1] and [19] for associative Banach algebras, hold true in Banach Jordan algebras. Their proofs, being natural extensions of the associative ones, are given here for the sake of completeness.

Theorem 8.

Let $\mathfrak{J}$ and $\mathcal{L}$ be two complex unital Banach Jordan algebras. Suppose $\mathrm{\Phi }:\mathfrak{J}⟼\mathfrak{L}$ is a linear, bijective, unital i.e $\mathrm{\Phi }\left(1_{\mathcal{J}}\right)=1_{\mathfrak{L}},$ isometric, that is ${\parallel \mathrm{\Phi }\left(x\right)\parallel }_{\mathcal{J}}={\parallel x\parallel }_{\mathcal{L}}$ and multiplicative, meaning $\mathrm{\Phi }\left(ab\right)=\mathrm{\Phi }\left(a\right)\mathrm{\Phi }\left(b\right)$ for all $a,b\in \mathcal{J}.$ Then

$${\mathrm{Sp}}_{ϵ}\left(a\right)={\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(a\right)\right)$$

for all $a\in \mathcal{J}$ and $ϵ>0.$

Proof. 

Let $a\in \mathcal{J}$ and $ϵ>0.$ If $\lambda \in {\mathrm{Sp}}_{ϵ}\left(a\right),$ then ${\parallel (\lambda \mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$ Since $\mathrm{\Phi }$ is an isometry,

$${\parallel \mathrm{\Phi }(\lambda \mathbf{1}-a)}^{-1}{\parallel =\parallel (\lambda -a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$$

Moreover, because $\mathrm{\Phi }$ is multiplicative and unital, inverses are preserved, hence

$$\mathrm{\Phi }\left(1_{\mathcal{J}}\right)=\mathrm{\Phi }\left(x{x}^{-1}\right)=\mathrm{\Phi }\left(x\right)\mathrm{\Phi }\left(x^{-1}\right)=1_{\mathcal{L}}.$$

Thus,

$${\parallel \left(\mathrm{\Phi }(\lambda \mathbf{1}-a)\right)}^{-1}{\parallel =\parallel \mathrm{\Phi }(\lambda \mathbf{1}-a)}^{-1}{\parallel =\parallel (\lambda \mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$$

Consequently, because $\mathrm{\Phi }$ is unital and linear,

$${\parallel (\lambda \mathbf{1}-\mathrm{\Phi }\left(a\right))}^{-1}{\parallel =\parallel (\mathrm{\Phi }(\lambda \mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$$

Hence $\lambda \in {\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(a\right)\right)$ and ${\mathrm{Sp}}_{ϵ}\left(a\right)\subseteq {\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(a\right)\right).$ By symmetry we can conversely show that ${\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(a\right)\right)\subseteq {\mathrm{Sp}}_{ϵ}\left(a\right),$ which completes the proof. □

Theorem 9.

Let $\mathfrak{J}$ and $\mathcal{L}$ be two complex unital Banach Jordan algebras and $ϵ>0.$ Let $\mathrm{\Phi }:\mathfrak{J}⟼\mathfrak{L}$ be a surjective ϵ-pseudospectrum preserving linear map. Then, Φ is spectra-preserving.

Proof. 

By assumption ${\mathrm{Sp}}_{ϵ}\left(a\right)={\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(a\right)\right),$ for all $a\in \mathcal{J}.$ Suppose $\lambda \notin \mathrm{Sp}\left(a\right),$ and choose ${t>ϵ\parallel (\lambda \mathbf{1}-a)}^{-1}\parallel .$ Then

$$\begin{array}{cc}\hfill {\parallel \left(t(\lambda \mathbf{1}-a)\right)}^{-1}\parallel & ={\displaystyle \frac{1}{t}}\parallel {(\lambda \mathbf{1}-a)}^{-1}\parallel \hfill \\ \hfill & <{\displaystyle \frac{1}{t}}{\displaystyle \frac{t}{ϵ}}={\displaystyle \frac{1}{ϵ}}.\hfill \end{array}$$

(11)

Hence,

$$\begin{array}{cc}\hfill t\lambda \notin {\mathrm{Sp}}_{ϵ}\left(ta\right)& ={\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(ta\right)\right)\hfill \\ & \supseteq \mathrm{Sp}\left(\mathrm{\Phi }\right(ta\left)\right)=t\mathrm{Sp}\left(\mathrm{\Phi }\right(a\left)\right),\hfill \end{array}$$

(12)

hence $\lambda \notin \mathrm{Sp}\left(\mathrm{\Phi }\right(a\left)\right).$ Therefore, $\mathrm{Sp}\left(\mathrm{\Phi }\right(a\left)\right)\subseteq \mathrm{Sp}\left(a\right).$ In a similar way, we can prove that $\mathrm{Sp}\left(a\right)\subseteq \mathrm{Sp}\left(\mathrm{\Phi }\right(a\left)\right).$ Hence $\mathrm{Sp}\left(\mathrm{\Phi }\right(a\left)\right)=\mathrm{Sp}\left(a\right).$ □

Theorem 10.

Let $\mathfrak{J}$ and $\mathcal{L}$ be two complex unital Banach Jordan algebras and $\mathrm{\Phi }:\mathfrak{J}⟼\mathfrak{L}$ be an ϵ-pseudospectrum preserving linear map for some $ϵ>0.$ Suppose Φ is multiplicative. Then, Φ preserves norms of all invertible elements of $\mathcal{J}.$

Proof. 

Suppose there exists an element $a\in \mathrm{Inv}\left(\mathcal{J}\right)$ such that $\parallel \mathrm{\Phi }\left(a^{-1}\right)\parallel \ne \parallel a^{-1}\parallel .$ Assume $\parallel a^{-1}\parallel <\parallel \mathrm{\Phi }\left(a^{-1}\right)\parallel ,$ choose $t>0$ such that

$$ϵ\parallel a^{-1}\parallel <t\le ϵ\parallel \mathrm{\Phi }\left(a^{-1}\right)\parallel =ϵ\parallel \mathrm{\Phi }{\left(a\right)}^{-1}\parallel .$$

Then ${\parallel \left(ta\right)}^{-1}\parallel <{\displaystyle \frac{1}{ϵ}}.$ Thus, $0\notin {\mathrm{Sp}}_{ϵ}\left(ta\right).$ But

$${\parallel \left(\mathrm{\Phi }\left(ta\right)\right)}^{-1}\parallel =\parallel {\displaystyle \frac{1}{t}}{\left(\mathrm{\Phi }\left(a\right)\right)}^{-1}\parallel ={\displaystyle \frac{1}{t}}\parallel \mathrm{\Phi }{\left(a\right)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$$

This implies $0\notin {\mathrm{Sp}}_{ϵ}\left(\mathrm{\Phi }\left(ta\right)\right),$ which contradicts the fact that $\mathrm{\Phi }$ preserves $ϵ$-pseudospectrum. □

6. Decomposition of Spectra and Pseudospectra

In a Banach algebra A, it is shown in Theorem 3.3 of [18], that there is a relationship between the spectrum and pseudospectrum of an element $a\in A$ with the spectra and pseudospectra of $pa=pap\in pAp$ and $qa=qaq\in qAq,$ where p is a non-trivial idempotent, that is, $p^2=p,p\ne 0,p\ne 1,$ and $q=1-p.$

Theorem 11.

Let A be a unital Banach algebra and let $p_1,\dots ,p_n\in A,$ such that $p_i^2=p_i$ for all i and ${\displaystyle \sum _{i=1}^n{p}_i=1.}$ Suppose $a{p}_i=p_{i}a$ for all $i.$ Let $ϵ>0$ and ${\displaystyle K=\underset{i}{max}\Vert p_i\Vert .}$ Then

$${\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{n}}}(A,a)\subseteq \bigcup _{i=1}^n{\mathrm{Sp}}_{ϵ}(p_{i}A{p}_i,p_{i}a)\subseteq {\mathrm{Sp}}_{Kϵ}(A,a).$$

Our aim is to extend this theorem to Banach Jordan algebras. At the present time, we are able only to obtain a decomposition of the pseudospectrum into the pseudospectra of the corresponding elements in the local Banach Jordan algebras, in the case of two orthogonal idempotents p and $q=1-p.$

Contrarily to the associative context, where $a{p}_i=p_{i}a$ was enough to prove for instance Theorem 12, in the non-associative situation we request that our elements operator commute. The following propositions can be found in [6].

Proposition 5.

Let $\mathfrak{J}$ be a Jordan algebra with identity, and let $p\in \mathfrak{J}$ be an idempotent element.Let $U_p\mathfrak{J}=\left\{U_{p}x\right|x\in \mathfrak{J}\}$ and ${\mathfrak{J}}_1\left(p\right)=\{x\in \mathfrak{J}|px=x\}.$ Then

(i) 

${\mathfrak{J}}_1\left(p\right)$ is a subalgebra of $\mathfrak{J},$ with identity $p.$

(ii) 

${\mathfrak{J}}_1\left(p\right)$ is semisimple and ${\mathfrak{J}}_1\left(p\right)=U_p\mathfrak{J}.$

(iii) 

If $x\in U_p\mathfrak{J},$ then ${\mathrm{Sp}}_{U_p\mathfrak{J}}\left(U_{p}x\right)\subset {\mathrm{Sp}}_{\mathfrak{J}}x.$

Proposition 6.

Let $a_1,\dots ,a_n\in \mathfrak{J}$ such that $a_i{a}_j=a_i^2{a}_j=a_i{a}_j^2=a_i^2{a}_j^2=0$ for $i\ne j.$ Then

$$\mathrm{Sp}(a_1+a_2+\dots +a_n)⫆\mathrm{Sp}\left(a_1\right)\cup \mathrm{Sp}\left(a_2\right)\cup \dots \mathrm{Sp}\left(a_n\right)\setminus \left\{0\right\}.$$

Corollary 2.

Let $\mathfrak{J}$ be a unital complex Banach Jordan algebra, let $p\in \mathfrak{J}$ an idempotent and let $q=1-p.$ Let $a\in \mathfrak{J}.$ Then

(i) 

$\mathrm{Sp}(\mathfrak{J},a)=\mathrm{Sp}(U_p\mathfrak{J},U_{p}a)\cup \mathrm{Sp}(U_q\mathfrak{J},U_{q}a),$

(ii) 

The spectral radius $\rho \left(a\right)=max\left\{\rho (U_p\mathfrak{J},U_{p}a),\rho (U_q\mathfrak{J},U_{q}a)\right\}.$

We recall that two elements $a,b$ in a Jordan algebra $\mathfrak{A}$ are said to operator commute if the operators $R_a,R_b$ commute, i.e. if $(a\circ c)\circ b=a\circ (c\circ b)$ for all $c\in \mathfrak{A},$ where ∘ is the Jordan product in $\mathfrak{A}.$ The next lemma is taken from [26] where it appears as Lemma 2.5.5 and it is proven on pages 46-47.

Lemma 1.

Let $\mathfrak{A}$ be a unital Jordan algebra and p an idempotent in $\mathfrak{A}.$ For any $a\in \mathfrak{A}$ the following conditions are equivalent:

(i) 

a and p operator commute,

(ii) 

$R_{p}a=U_{p}a,$

(iii) 

$a=(U_p+U_{p\perp })a,$

(iv) 

a and p generate an associative subalgebra of $\mathfrak{A}.$

Moreover, $U_p\mathfrak{A}$ and $U_{p\perp }\mathfrak{A}$ are subalgebras of $\mathfrak{A},$ and $a\circ b=0$ if $a\in U_p\mathfrak{A}$ and $b\in U_{p\perp }\mathfrak{A}.$

Now, suppose p is a central projection in $\mathfrak{A},$ that is p operator commutes with every other element of $\mathfrak{A}.$ Then by Lemma 1, one can write $J=U_{p}J\oplus U_{p\perp }J.$

Theorem 12.

Let $\mathfrak{J}$ be a unital complex Banach Jordan algebra, let $p\in \mathfrak{J}$ be a central projection and $p^{\perp }=1-p.$ Let $ϵ>0$ and $K=max\{\parallel p\parallel ,\parallel 1-p\parallel \}.$ Then

$${\mathrm{Sp}}_{ϵ}(J,a)\subseteq {\mathrm{Sp}}_{ϵ}(U_{p}J,U_{p}a)\bigcup {\mathrm{Sp}}_{ϵ}(U_{p^{\perp }}J,U_{p^{\perp }}a)\subseteq {\mathrm{Sp}}_{Kϵ}(\mathfrak{J},a).$$

Proof. 

Let $ϵ>0$ and

$${\displaystyle \lambda \in {\mathrm{Sp}}_{ϵ}(U_p\mathfrak{J},U_{p}a)\bigcup {\mathrm{Sp}}_{ϵ}(U_{p^{\perp }}\mathfrak{J},U_{p^{\perp }}a)).}$$

If $\lambda \in \mathrm{Sp}(U_p\mathfrak{J},U_{p}a),$ then $\lambda \in \mathrm{Sp}(\mathfrak{J},a)\subseteq {\mathrm{Sp}}_{ϵ}(\mathfrak{J},a)\subseteq {\mathrm{Sp}}_{Kϵ}(\mathfrak{J},\mathfrak{a}),$ by Proposition 5(iii).

On the other hand, if $\lambda ·p-U_{p}a$ is invertible, then $\lambda ·\mathbf{1}-a$ is invertible by Corollary 2, then it must be the case that $\lambda \in {\mathrm{Sp}}_{ϵ}(U_{p}J,U_{p}a).$ Then $\parallel {(\lambda ·p-U_{p}a)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}.$ Since p is a projection, then $U_{p}a=pa,$ which leads to

$$\parallel {(\lambda ·p-U_{p}a)}^{-1}\parallel = \parallel {\left(U_p(\lambda ·\mathbf{1}-a)\right)}^{-1}\parallel = \parallel {\left(p(\lambda ·\mathbf{1}-a)\right)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}$$

Now, exactly as in the proof of Theorem 3.3 in [18], this yields a bound for the norm of ${(\lambda ·\mathbf{1}-a)}^{-1}$ as follows

$${\parallel \left(p(\lambda ·\mathbf{1}-a)\right)}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}\Rightarrow \parallel {(\lambda ·\mathbf{1}-a)}^{-1}\parallel \ge {\displaystyle \frac{1}{\parallel p\parallel ϵ}}>{\displaystyle \frac{1}{Kϵ}}$$

Thus,

$${\displaystyle \lambda \in {\mathrm{Sp}}_{ϵ}(U_p\mathfrak{J},U_{p}a)\bigcup {\mathrm{Sp}}_{ϵ}(U_{p^{\perp }}\mathfrak{J},U_{p^{\perp }}a))\subseteq {\mathrm{Sp}}_{Kϵ}(\mathfrak{J},a).}$$

Next, suppose

$${\displaystyle \lambda \notin {\mathrm{Sp}}_{ϵ}(U_p\mathfrak{J},U_{p}a)\bigcup {\mathrm{Sp}}_{ϵ}(U_{p^{\perp }}\mathfrak{J},U_{p^{\perp }}a)).}$$

Then $\lambda ·p-U_{p}a$ is invertible with inverse $U_{p}b,$ for some $b\in \mathfrak{J},$ and $\lambda ·p^{\perp }-U_{p^{\perp }}a$ is invertible with inverse $U_{p^{\perp }}c,$ for some $c\in \mathfrak{J}.$ Moreover, $\parallel U_{p}b\parallel <{\displaystyle \frac{1}{ϵ}}$ and $\parallel U_{p^{\perp }}c\parallel <{\displaystyle \frac{1}{ϵ}}.$ Then $\lambda \mathbf{1}-a$ is invertible (by Lemma 2) with inverse $U_{p}b+U_{p^{\perp }}c$ (by Lemma 1 (iv)) and further, ${\displaystyle \parallel {\left(U_p(\lambda \mathbf{1}-a)\right)}^{-1}\parallel =\parallel U_{p}b+U_{p^{\perp }}c\parallel <2\frac{1}{ϵ}.}$ Hence we conclude that

$${\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{2}}}(J,a)\subseteq {\mathrm{Sp}}_{ϵ}(U_{p}J,U_{p}a)\bigcup {\mathrm{Sp}}_{ϵ}(U_{p^{\perp }}J,U_{p^{\perp }}a)\subseteq {\mathrm{Sp}}_{Kϵ}(\mathfrak{J},a).$$

□

Corollary 3.

Let $\mathfrak{J}$ be a unital complex Banach Jordan algebra, let $p\in \mathfrak{J}$ an idempotent and let $q=\mathbf{1}-p.$ Let $a\in \mathfrak{J}$ and suppose that $\parallel p\parallel =\parallel q\parallel =1.$ Then

$${\mathrm{Sp}}_{{\displaystyle \frac{ϵ}{2}}}(J,a)\subseteq {\mathrm{Sp}}_{ϵ}(U_{p}J,U_{p}a)\cup {\mathrm{Sp}}_{ϵ}(U_{q}J,U_{q}a)\subseteq {\mathrm{Sp}}_{ϵ}(\mathfrak{J},a).$$

Exploring pseudospectra in Banach Jordan algebras algebras is a novel and fertile area of research, bridging non-associative algebra, spectral theory, and stability analysis. There are many interesting questions to answer in this regard, for instance it will be of great importance to obtain a theorem for a class of Banach Jordan algebras similar to Theorem 12.

7. Roch-Silberman Theorem in a JB-Algebra

First, we recall the well-known theorem of Roch and Silberman established in [23] for $C^{*}$-algebras. Indeed, let $ϵ>0$ and let $\mathcal{B}$ a $C^{*}$-algebra with identity. The $ϵ$-pseudospectrum ${\mathrm{Sp}}_{ϵ}\left(b\right)$ of an element $b\in \mathcal{B}$ is the set

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(b\right)=\left\{{t\in \mathbb{C};b-t·\mathbf{1}\mathrm{is}\mathrm{not}\mathrm{invertible}\mathrm{or}\parallel (b-t·\mathbf{1})}^{-1}\parallel \ge {\displaystyle \frac{1}{ϵ}}\right\}.\end{array}$$

(13)

Here is an equivalent characterization of the $ϵ$-pseudospectrum established as Proposition 4.15 in [23] for $C^{*}$-algebras.

Theorem 13.

The ϵ-pseudospectrum ${\mathrm{Sp}}_{ϵ}\left(b\right)$ of an element $b\in \mathcal{B}$ coincides with the set

$$\begin{array}{c}\hfill \left\{t\in \mathbb{C};thereisap\in \mathcal{B}with\parallel p\parallel \le ϵandt\in \mathrm{Sp}(b+p)\right\}.\end{array}$$

(14)

Our aim is to generalize this theorem to a particular class of Banach Jordan algebras, called $J{B}^{*}$-algebras. A complex Banach Jordan algebra $\mathcal{J}$ with isometric involution * is called a $J{B}^{*}$-algebra if $\parallel {\mathrm{U}}_x{x}^{*}{\parallel =\parallel x\parallel }^3$ for all $x\in J.$ For instance, every $C^{*}$-algebra is a $J{B}^{*}$-algebra when equipped with its natural Jordan product, the original norm and involution. The concepts of $JB$-algebras and $J{B}^{*}$-algebras are essentially equivalent as shown in [29]. More precisely, the self-adjoint part ${\mathcal{J}}_s=\{x\in \mathcal{J}:x^{*}=x\}$ of a $J{B}^{*}$-algebra $\mathcal{J}$ is a $JB$-algebra under the retricted norm whereas the complexification ${\mathcal{J}}_c=\mathcal{J}\otimes \mathbb{C}$ of a $JB$-algebra $\mathcal{J}$ is a $J{B}^{*}$-algebra under a unique norm extending the $JB$-algebra norm on $\mathcal{J}.$

Theorem 14.

(a) If $\mathcal{A}$ is a $J{B}^{*}$-algebra then the set of self-adjoint elements of $\mathcal{A}$ is a $JB$-algebra.

(b) If $\mathcal{B}$ is a $JB$-algebra then under a suitable norm the complexification ${\mathcal{C}}_{\mathcal{B}}$ of $\mathcal{B}$ is a $J{B}^{*}$-algebra.

The basic examples $JB$-algebras are of the following types. Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{B}\left(\mathcal{H}\right)$ the Banach algebra of bounded linear operators on $\mathcal{H}.$

(a)

The real Banach space $\mathcal{B}\left(\mathcal{H}\right)$ of all bounded self-adjoint operators on $\mathcal{H}$ is a $JB$-algebra.

(b)

Any closed unital subalgebra $\mathcal{S}$ of self-adjoint operators of $\mathcal{B}\left(\mathcal{H}\right),$ is called a $JC$-algebra. Every $JC$-algebra is a special $JB$-algebra. The converse is also true by the structure theory of [2].

For an element a in a unital JB-algebra $\mathfrak{J},$ the spectrum $\mathrm{Sp}(\mathfrak{J},a)$ is a nonempty and compact subset of $\mathbb{R}$ and coincides with $\mathrm{Sp}\left(\mathcal{C}\right(a),a),$ thus the spectrum does not vary with the closed subalgebra of $\mathfrak{J}$ containing a and the unit element.

The main result of Roch-Silbermann is stated and proved in the general setting of $C^{*}$-algebras which are a proper subclass of the larger category of $J{B}^{*}$-algebras. These constitute nowadays an important field of study as can be seen in [10] and the references cited within pertaining to Quantum Machanics. Similarly to JB-algebras, the concepts of invertibility and spectrum are well defined in $J{B}^{*}$-algebras. We recall that the Shirshov-Cohn theorem states that a Jordan algebra, unital or not, with two generators is special. As a consequence, a JB-algebra generated by two elements is a JC-algebra (see [3], Theorem 7.2.5). Hence, any $J{B}^{*}$-subalgebra of a $J{B}^{*}$-algebra, that is generated by two self-adjoint elements and the identity, is a $J{C}^{*}$-algebra, that is, a $J{B}^{*}$-subalgebra of some $\mathcal{B}\left(\mathcal{H}\right)$ for some Hilbert space $\mathcal{H}$ [29], Corollary 2.2. We prove the second inclusion of Proposition 4.15 [23], for self-adjoint elements; in this case, the proof of Roch-Silberman works verbatim because the subalgebra generated by $1,a,a^{*}$ is associative.

Proposition 7.

For self-adjoint element a in a unital JB*-algebra $\mathfrak{J},$ the pseudospectrum

$${\mathrm{Sp}}_{ϵ}\left(a\right)\subseteq \{t\in \mathbb{C}:\exists p\in J\mathit{with}|\left|p\right||\le ϵ\mathit{and}t\in \mathrm{Sp}(a+b)\}$$

Proof. 

Same as the proof of Proposition 4.15 in [23], since the subalgebra generated by a self-sdjoint element and the identity is associative by Proposition 3.41 in [16].

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Final comments

Is it possible to get a more general result than Proposition 7? In other words, does the following theorem hold in Banach Jordan algebras?

Theorem 15.

For an element a in a unital $J{B}^{*}$-algebra $\mathfrak{J},$ the pseudospectrum

$$\begin{array}{c}\hfill {\mathrm{Sp}}_{ϵ}\left(a\right)=\bigcup _{b\in \mathfrak{J},\parallel b\parallel \le ϵ}\mathrm{Sp}(a+b)\end{array}$$