On the Method for Proving RH Using the Alcantara-Bode Equivalence

1. Introduction

The author of [2] proved that the integral operator of interest is Hilbert-Schmidt. From the Hilbert Schmidt operators that are linear and bounded we need their compactness property useful for us considering operator convergent approximations on finite dimension subspaces. We extend the research to linear bounded operators that are included in this class.

The functional-numerical method is built on the result obtained with Theorem 1: if a linear bounded operator T on a separable Hilbert space H is strict positive on a dense set S, meaning that if there is a constant $\alpha >0$ such that $\langle Tu,u\rangle \ge {\alpha \parallel u\parallel }^2$ for any $u\in S$, then T is injective - equivalently, its null space contains only element 0: $N_T=\left\{0\right\}$. When the dense set is an infinite union of finite dimension subspaces from a family F, $S={\cup }_{n=1,\infty }{S}_n$, then the strict positivity on each subspace $S_n$ will attract the positivity on the dense set, $\langle Tu,u\rangle >0$ for every $u\in S$. This property ensures that the operator has no zeros in the dense set. For investigation of the zeros from outside of the dense set we need more properties to be satisfied. We will name such elements as being ’eligible’ zeros, and denote $E:=H\setminus S$ the difference set or set of eligible zeros.

Such subspaces can be built by discretization methods based on domain decomposition such as those used in finite element or Galerkin methods ([3]). Instead of considering on each subspace the restriction of the operator (see criterion in Note 1), we use their operator approximation for which the discretization schema ensures their convergence. On convergent approximations, the positivity property ensures that the operator has not zeros in the dense set (Observation 2). A bound of the positivity parameters ensures that there are no zeros in the difference set too, so the operator null space contains only zero.

In our analysis we consider only eligible elements from the unit sphere ($\parallel u\parallel =1$), because for any $u\in H$ not null, both u and $u/\parallel u\parallel$ are together or are not together in the null space of T. When necessary we use the associated Hermitian operator ($T^{*}T$) that is non negative on the whole space ($\langle T^{*}Tv,v\rangle =\langle Tv,Tv\rangle \ge 0$ on H) and has the same null space as T.

2. Injectivity Criteria

Let H be a separable Hilbert space and denote with $\mathcal{L}\left(H\right)$ the class of linear bounded operators on H. The following theorem is used as support for the methods using finite rank operator approximations on subspaces built on a multilevel structure whose union is dense.

Theorem 1. 

If $T\in \mathcal{L}\left(H\right)$ is strict positive on a dense set of a separable Hilbert space then $N_T=\left\{0\right\}$, equivalently T is injective.

Proof. 

The set $S\subset H$ is dense if its closure coincides with H. Then, if $w\in E:=H\setminus S$, for every $\epsilon >0$ there exists $u_{\epsilon ,w}\in S$ such that $\parallel w-u_{\epsilon ,w}\parallel <\epsilon$.

Now, (1) below results as follows. If $\parallel w\parallel \ge \parallel u_{\epsilon ,w}\parallel$:

$0\le \parallel w\parallel -\parallel u_{\epsilon ,w}\parallel =\parallel w-u_{\epsilon ,w}+u_{\epsilon ,w}\parallel -\parallel u_{\epsilon ,w}\parallel \le \parallel w-u_{\epsilon ,w}\parallel <\epsilon$.

If $\parallel u_{\epsilon ,w}\parallel \ge \parallel w\parallel$ instead, then:

$0\le \parallel u_{\epsilon ,w}\parallel -\parallel w\parallel =\parallel u_{\epsilon ,w}-w+w\parallel -\parallel w\parallel \le \parallel w-u_{\epsilon ,w}\parallel <\epsilon$.

Therefore, given $w\in E$, for every $\epsilon >0$ there exists $u_{\epsilon ,w}\in S$ such that

$$|\parallel w\parallel -\parallel u_{\epsilon ,w}\parallel |<\epsilon$$

(1)

Let w be an eligible element from the unit sphere, $\parallel w\parallel =1$ and take ${\epsilon }_n=1/n$. Then there exists at least one element $u_{{\epsilon }_n,w}\in S$ such that $\parallel u_{{\epsilon }_n,w}-w\parallel <{\epsilon }_n$ holds. From (1), ∣ 1 - $\parallel u_{{\epsilon }_n,w}\parallel \mid$$<1/n$ showing that, for any choices of a sequence approximating w, $u_{{\epsilon }_n,w}\in S,n\ge 1$, it verifies $\parallel u_{{\epsilon }_n,w}\parallel \to 1$.

Let $T\in \mathcal{L}\left(H\right)$ strict positive on S, i.e. $\exists \alpha >0$ such that $\forall u\in S$, $\langle Tu,u\rangle \ge {\alpha \parallel u\parallel }^2$.

If $\exists w\in E\cap N_T,\parallel w\parallel =1$, consider a sequence of approximations of w, $u_{{\epsilon }_n,w}\in S,n\ge 1$ that, as we showed, has its normed sequence converging to 1. From the positivity of T on the dense set S, follows:

$$\alpha \parallel u_{{\epsilon }_n,w}{\parallel }^2\le \langle T{u}_{{\epsilon }_n,w},u_{{\epsilon }_n,w}\rangle =\langle T(u_{{\epsilon }_n,w}-w),u_{{\epsilon }_n,w}\rangle <{\epsilon }_n\parallel T\parallel \parallel u_{{\epsilon }_n,w}\parallel$$

(2)

With c=$\parallel T\parallel /\alpha$, we obtain $\parallel u_{{\epsilon }_n,w}\parallel \le c/n$. Subsequently, $\parallel u_{{\epsilon }_n,w}\parallel \to 0$ with $n\to \infty$, contradicting its convergence $\parallel u_{{\epsilon }_n,w}\parallel \to 1$ with $n\to \infty$.

This occurs for any choice of sequence of approximations of w, verifying

$\parallel w-u_{{\epsilon }_n,w}\parallel <{\epsilon }_n,n\ge 1$, when $Tw=0$.

Thus $w\notin N_T$, valid for any $w\in E,\parallel w\parallel =1$, proving the theorem because no zeros of T there are in S either. □

We use this generic theorem for providing a method for investigating the injectivity by using operator approximation positivity properties on finite dimension subspaces whose union is a dense set.

2.1. Setting the Environment

Suppose that there exists a dense set S as a result of an union of finite dimension of including subspaces of a family F: $S_n\subset S_{n+1},n\ge 1$, $S={\cup }_{n\ge 1}{S}_n,\overline{S}=H$. Given $T\in \mathcal{L}\left(H\right)$, let $\{T_n;n\ge 1\}$ be a sequence of its operator approximations on the family F having the following properties:

i) ${ϵ}_n:=\parallel T-T_n\parallel \to 0$for$n\to \infty$; (convergence of approximations)

ii) $\langle T_{n}v,v\rangle \ge {\alpha }_n{\parallel v\parallel }^2$, ${\alpha }_n>0$$\forall v\in S_n,S_n\in F$ (positivity of approximations).

For $T\in \mathcal{L}\left(H\right)$ where $H=L^2(\Omega )$ with $\Omega \subset \mathcal{R}$ the discretization schema for T defined in the next paragraph ensures its operator approximations convergence. Then, we need to deal with its positivity on subspaces in the chosen dense set.

In the hypotheses i), ii) the following results hold.

Observation 1. Let ${\beta }_n\left(u\right):=\parallel u-u_n\parallel$ be the normed residuum of the element $u\in E$ after its orthogonal projection onto $S_n$. Thus ${\beta }_n\left(u\right)\to 0$ for $n\to \infty$.

Proof. 

For $ϵ>0$, from the density of the set S in H there exists $u_{ϵ}\in S$ verifying $\parallel u-u_{ϵ}\parallel <ϵ$, as per the observations made in the proof of the Theorem 1. Let $S_{n_{ϵ}}$ be the coarsest subspace, i.e. with the smallest dimension, from the family of subspaces containing $u_{ϵ}$. Because the best approximation of u in $S_{n_{ϵ}}$ is its orthogonal projection, we obtain

${\beta }_{n_{ϵ}}\left(u\right):=\parallel u-P_{n_{ϵ}}u\parallel \le \parallel u-u_{ϵ}\parallel <ϵ$, valid for every $ϵ>0$, proving our assertion. Rewriting,

${\beta }_n\left(u\right):=\parallel u-P_{n}u\parallel =\parallel (I-P_n)u\parallel \le \parallel I-P_n\parallel \parallel u\parallel \to 0$ for $n\to \infty$ for any $u\in H$ with $P_n$ being the orthogonal projection onto $S_n,\forall n\ge 1$. □ □

Theorem 2. 

If T is positive on S, $\langle Tu,u\rangle >0$ for any $u\in S$ and the set $\{{\alpha }_n,n\ge 1\}$ of positivity parameters is bounded, i.e. there exists $\alpha >0$ such that

iii) ${\alpha }_n\ge \alpha >0,n\ge 1$,then$N_T=\left\{0\right\}$.

Proof. 

Being positive on S, the operator has no zeros from the dense set. Remembering that any $u\notin S$ is an eligible one, we will show now that the operator does not have any eligible zeros too.

For a not null $u\in E:=H\setminus S$, $\parallel u\parallel =1$ and $u_n:=P_{n}u$ its non null orthogonal projection over a coarser subspace $S_n$, $n\ge n_0:=n_0\left(u\right)$, from (ii),

$$0<{\alpha }_n\parallel u_n{\parallel }^2\le \langle T_n{u}_n,u_n\rangle \le \parallel T_n{u}_n\parallel \parallel u_n\parallel .$$

(3)

If $u\in N_T$, then the estimation of $\parallel T_n{u}_n\parallel$ becomes:

$\parallel T_n{u}_n\parallel =\parallel (T_n{u}_n-T{u}_n)+(T{u}_n-Tu)\parallel$$\le (\parallel T-T_n\parallel \parallel u_n\parallel +\parallel T\parallel \parallel u-u_n\parallel )$

= $({ϵ}_n\parallel u_n\parallel +\parallel T\parallel {\beta }_n)$ Because $\parallel u_n\parallel \to 1$, ${ϵ}_n\to 0$ (from i)) and ${\beta }_n:={\beta }_n\left(u\right)\to 0$ (from Observation 1), then $\parallel T_n{u}_n\parallel \to 0$. From iii) and (3),

$\alpha \parallel u_n{\parallel }^2\le ({ϵ}_n+\parallel T\parallel {\beta }_n/\parallel u_n\parallel )\parallel u_n{\parallel }^2$.

From 1 = ${\parallel u\parallel }^2={\parallel u_n\parallel }^2+{\beta }_n^2$, results ${\beta }_n/\parallel u_n\parallel ={\beta }_n/\sqrt{1-{\beta }_n^2}\to 0$. So,

$0<\alpha \le \left({ϵ}_n+\parallel T\parallel {\beta }_n/\parallel u_n\parallel \right)\to 0$.

The inequality is violated from a range $n_1\ge n_0$, involving $u\notin N_T$. This conclusion is valid for any supposedly zero of T in E. Because T has no zeros in the eligible set and it has no zeros in the dense set too, we obtain $N_T=\left\{0\right\}$. □

The previous theorem is a formulation of Theorem 1 on the family of finite dimension subspaces whose union is a dense set, still having to deal with the operator positivity on that dense set. The following observation moves the investigation on the approximation subspaces.

Observation 2. If T verifies the approximation properties then $S\cap N_T=\left\{0\right\}$.

Proof. 

If there exists $u\in S\cap N_T$ then there exists a coarser subspace $S_{n_0}$ such that for $n\ge n_0$$u\in S_n$ and $Tu=0$. Then, on $S_n$ we have:

${0<\alpha \parallel u\parallel }^2\le {\alpha }_n{\parallel u\parallel }^2\le \langle T_{n}u,u\rangle =\langle (T_n-T)u,u\rangle \le {ϵ}_n{\parallel u\parallel }^2$, meaning ${ϵ}_n\ge \alpha$ where ${ϵ}_n\to 0$ and $\alpha$ is a positive not null constant. This is a contradiction showing that $S\cap N_T=\left\{0\right\}$. □ □

Theorem 3. 

If $T\in \mathcal{L}\left(H\right)$ verifies the approximation properties i), ii) and the set $\{{\alpha }_n,n\ge 1\}$ of positivity parameters is bounded, i.e. there exists $\alpha >0$ such that

iii) ${\alpha }_n\ge \alpha >0,n\ge 1$,then$N_T=\left\{0\right\}$, equivalently Tis injective.

Proof. 

In the hypothesis i) and ii) the operator has no zeros in S (from Observation 2). For showing that the operator has no zeros in the eligible set, we could proceed like in previous theorem, obtaining a contradiction in the relationship for any $u\in E$ supposed to verify $Tu=0$:

$0<\alpha \le \left({ϵ}_n+\parallel T\parallel {\beta }_n/\parallel u_n\parallel \right)$ where ${ϵ}_n\to 0$ (from i)), and ${\beta }_n\to 0$ from Observation 1.

Then, $N_T=\left\{0\right\}$. □

□

Theorem 4. 

If T is Hermitian verifying the approximation properties and the set $\{{\alpha }_n,n\ge 1\}$ of the positivity parameters is bounded, i.e. there exists $\alpha >0$ such that

iii) ${\alpha }_n\ge \alpha >0,n\ge 1$,then T is strictly positive on the dense set.

Proof. 

From the convergence to zero of the sequence ${ϵ}_n,n\ge 1$ there exists a parameter ${ϵ}_0$ such that ${ϵ}_0:=ma{x}_n\{{ϵ}_n;{ϵ}_n<\alpha \}$, corresponding to a subspace $S_{n_0},n_0<\infty$. This parameter is independent of any $v\in S$ and, because of the inclusion property, for any $n<n_0$ we have $S_n\subset S_{n_0}$. We could consider $S_{n_0}$ to be $S_1$ discarding a finite number of subspaces or, we could consider v to be inside of $S_{n_0}$. Then:

${\alpha }_n\ge \alpha >{ϵ}_0\ge {ϵ}_n$ for $n\ge 1$, resulting $({\alpha }_n-{ϵ}_n)>(\alpha -{ϵ}_0)>0$$\forall n\ge 1$.

For an arbitrary $v\in S$ there exists a coarser subspace (i.e. with a smaller dimension) $S_n,n\ge n_1:=n_1\left(v\right)$, for which $v\in S_n$. For it, amending the inequality in (3) to the semi-positivity property of the Hermitian operator yields:

$0\le \langle Tv,v\rangle =\langle T_{n}v,v\rangle +\langle (T-T_n)v,v\rangle$.

Because $T_n$ is positive on $S_n$ and T is semi-positive, $T_n-T$ or $T-T_n$ is positive for that v and, the second inner product in the right side of the equality is real valued: $\langle (T_n-T)v,v\rangle \ge 0$ or, $\langle (T_n-T)v,v\rangle <0$.

If $\langle (T_n-T)v,v\rangle \ge 0$, then $0\le \langle (T_n-T)v,v\rangle \le {ϵ}_n{\parallel v\parallel }^2$ and,

$\langle T_{n}v,v\rangle -\langle (T_n-T)v,v\rangle \ge \langle T_{n}v,v\rangle -{ϵ}_n{\parallel v\parallel }^2\ge {\alpha }_n{\parallel v\parallel }^2-{ϵ}_n{\parallel v\parallel }^2>(\alpha -{ϵ}_0){\parallel v\parallel }^2$.

Follows $\langle Tv,v\rangle >(\alpha -{ϵ}_0){\parallel v\parallel }^2$.

If instead $\langle (T_n-T)v,v\rangle <0$, then $\langle Tv,v\rangle$ is a sum of two positive quantities so, greater than any of them:

$\langle Tv,v\rangle \ge {\alpha }_n{\parallel v\parallel }^2\ge {\alpha \parallel v\parallel }^2>(\alpha -{ϵ}_0){\parallel v\parallel }^2$.

Thus, taking $\alpha \left(T\right)=(\alpha -{ϵ}_0)>0$, ${\langle Tv,v\rangle >\alpha \left(T\right)\parallel v\parallel }^2$ holds $\forall v\in S$.

So, T is strict positive on the dense set. From Theorem 1, $N_T=\left\{0\right\}$. □

Corollary 1. 

Let $T\in \mathcal{L}\left(H\right)$. If the sequence of operator approximations of its associated Hermitian operator $Q:=T^{*}T$ verifies:

$\langle Q_{n}v,v\rangle \ge {\alpha }_n{\parallel v\parallel }^2\ge \alpha {\parallel v\parallel }^2,\forall v\in S_n$,on any$S_n\in F$,where$\alpha >0$is a constant,

then Q is strictly positive on S and, $N_T=N_Q=\left\{0\right\}$.

Proof. 

Suppose that T and Q operator approximations are convergent, verifying both i) property. The associated Hermitian operator is also non negative on the subspaces of F. We are under the hypotheses of the Theorem 3, both positivity and iii) properties hold, showing that the operator is strictly positive on the dense set. According to Theorem 1, $N_Q=\left\{0\right\}$. Because $N_T=N_Q$, we obtain $N_T=\left\{0\right\}$. □

Note 1. (A Criterion for operator restrictions.)Let $T\in \mathcal{L}\left(H\right)$ be positive on the subspaces $S_n,n\ge 1$ whose union S is a dense set, verifying: $\langle Tv,v\rangle \ge {\alpha }_n{\parallel v\parallel }^2$ for every $v\in S_n$, where ${\alpha }_n\to 0$ with $n\to \infty$. Consider now the parameters:

${\mu }_n:={\alpha }_n\left(T\right)/{\omega }_n$where${\omega }_n$verifies$\parallel T^{*}v\parallel \le {\omega }_n\parallel v\parallel ,\forall v\in S_n,n\ge 1$.

If T is Hermitian, ${\alpha }_n\left(T\right)=mi{n}_j\left({\lambda }_{{T|}_{S_n}}^j\right)$ and ${\omega }_n\left(T\right)=ma{x}_j\left({\lambda }_{{T|}_{S_n}}^j\right)$ where$\left\{{\lambda }_{{T|}_{S_n}}^j\right\}$j=1,n are eigenvalues of the restriction of T on$S_n$.

If there exists a constant $C>0$such that${\mu }_n\ge C$for every$n\ge 1$, then$N_T=\left\{0\right\}$.

Proof. 

Suppose that there exists $u\in (H\setminus S)\cap N_T$, $\parallel u\parallel =1$ and let $u_n$ its orthogonal projection on $S_n,n\ge 1$. Then, from the (strict) positivity of T on each of the subspaces $S_n,n\ge 1$ (see (2)):

${\alpha }_n\left(T\right)\parallel u_n{\parallel }^2\le \langle T{u}_n,u_n\rangle =\langle T(u_n-u),u_n\rangle =\langle (u_n-u),T^{*}{u}_n\rangle \le {\beta }_n{\omega }_n\parallel u_n\parallel$

Then, from

$C\le {\mu }_n\le {\beta }_n/\sqrt{1-{\beta }_n^2}\to 0$ where ${\beta }_n:={\beta }_n\left(u\right)=\parallel u-u_n\parallel$, we obtain a contradiction. Thus, $u\notin N_T$ for any $u\in H\setminus S$. Follows: $N_T=\left\{0\right\}$ because there are no zeros from S too. □

3. Integral Operator Approximations

Let $H:=L^2(0,1)$. The semi-open intervals of equal lengths $h=2^{-m},m\in N$, nh = 1, ${\Delta }_{h,k}=((k-1)/2^m,k/2^m]$, $k=1,n-1$ together with the open ${\Delta }_{h,n}$ are defining for $m\ge 1$ a partition of (0,1), k=1,n, $n=2^m,nh=1$. As in [5], to build the subspaces with their union a dense set, consider the interval indicator functions with the supports of these intervals (k=1,n):

$$I_{h,k}\left(t\right)=1\mathrm{for}t\in {\Delta }_{h,k}\mathrm{and}0\mathrm{otherwise}$$

(4)

Consider now, the family F of finite dimensional subspaces $S_h$ that are the linear spans of interval indicator functions of the h-partitions defined by (4) with disjoint supports, i.e. pairwise disjoint, $S_h=span\{I_{h,k};k=1,n,nh=1\}$, built on a multi-level structure and denote $S:={\cup }_{n\ge 2}{S}_h,nh=1$, their union that is a dense set in H well known in literature. As an exercise in [9], the author shows the density of step functions built on functions of indicators of intervals pairwise disjoint in $L^p(\Omega ),1\le p<\infty$ where $\Omega$ is a domain in $\mathcal{R}$. In [5] and [6] the authors used the dense set built on (4) to obtain the best rate of convergence to zero of the eigenvalues of integral operators with Mercer kernels ([8]).

Note 2.

a) The family F is an including family of subspaces;

b) S is dense and, $\forall u\in E$${\beta }_h\left(u\right):=\parallel u-P_{h}u\parallel \to 0$ with $n\to \infty ,nh=1$.

c) For any $u\in L^2(0,1)$ the functions $\left(u{I}_{h,k}\right)$, $\left(I_{h,k}u\right)$ not necessarily in S have the support ${\Delta }_{h,k}$: $\left(u{I}_{h,k}\right)\left(t\right)=\left(I_{h,k}u\right)\left(t\right)=0$ for $t\notin {\Delta }_{h,k}$).

Proof. 

a). The including property $S_h\subset S_{h/2},n\ge 2,nh=1$ is obtained from (4) by halving the mesh h, observing that any $I_{h,i}\in S_h,i=1,n$ could be embed into $S_{h/2}$ as per:

$I_{h,i}=I_{h/2,2i-1}+I_{h/2,2i}\in S_{h/2}$.

b). For any $u\in H\setminus S,u\ne 0$, from an index $n_0,n_0{h}_0=1$ its orthogonal projection $P_{h}u=h^{-1}{\sum }_{k=1,n}{I}_{h,k}{c}_{h,k}\in S_h$, $P_{h}u\ne 0,nh=1$. Here, $c_{h,k}=\langle u,I_{h,k}\rangle$. From a) we have $P_{h}u\in S_{h/2}$. So, ${\beta }_{h/2}\le {\beta }_h$ because the best approximation of u in $S_{h/2}$ is given by $P_{h/2}u$. Therefore, ${\beta }_h$ is decreasing with h decreasing. We could remark that $\parallel P_{h}u-u\parallel \to 0$ for any $u\in E$ (equivalently with: given $ϵ>0$ there exists h such that ${\beta }_h\left(u\right)<ϵ$), holds iff S is dense.

c). From (4), $\left(u{I}_{h,k}\right)$ is the restriction of u to ${\Delta }_{h,k}$, so we have $\left(u{I}_{h,k}\right)\left(t\right)=0$ for $t\notin {\Delta }_{h,k}$. □

3.1. Convergent Operator Approximations

Choosing open subintervals or closed subintervals for partitioning the domain, the unions of the subspaces generated in both cases are also dense, ([1]): if one from three sets is dense, then the other two sets are dense. The option for semi-open subintervals partitions ensures that the subspaces including property and, every pair of indicator subinterval functions has disjoint supports obtaining as result sparse diagonal matrix representations.

Citing [5], (pg 986), the integral operator $P_h^r,n\ge 1$ with the kernel function:

$$r_h(y,x)=h^{-1}\sum _{k=1,n}{I}_{h,k}\left(y\right)I_{h,k}\left(x\right)$$

(5)

is a finite rank integral operator orthogonal projection having the spectrum {0, 1} with the eigenvalue 1 of the multiplicity n (nh=1) corresponding to the orthogonal eigenfunctions $I_{h,k},k=1,n$. Then, $\forall u\in H$, $P_h^{r}u\in S_h$ and, ${\left(P_h^r\right)}^2=P_h^r$ for $n\ge 2,nh=1$. For any $u\in H$,

$P_h^{r}u={\int }_0^1{r}_{h}u\left(x\right)dx=h^{-1}{\sum }_{k=1,n}{c}_k{I}_{h,k}\in S_h$, the constants $c_k$, k=1,n being given by

$c_k={\int }_0^{1}u\left(x\right)I_{h,k}\left(x\right)dx:=\langle u,I_{h,k}\rangle$. Thus, $P_h^r$ is an orthogonal projection onto $S_h$ like is mentioned in [5].

Let $T_{\rho }\in \mathcal{L}\left(H\right)$. Its integral operator approximation on $S_h$ denoted by $T_{{\rho }_h}$ is a finite rank operator approximation, with the kernel function (citing again [5]):

$${\rho }_h(y,x)=h^{-1}\sum _{k=1,n}{I}_{h,k}\left(y\right)\rho (y,x)I_{h,k}\left(x\right):=h^{-1}\sum _{k=1,n}{\rho }_h^k(y,x)$$

(6)

where the pieces ${\rho }_h^k,k=1,n$ of the kernel function ${\rho }_h$ in the sum have disjoint supports in $L^2{(0,1)}^2$, namely ${\Delta }_{h,k}^2,k=1,n,nh=1$. The operator approximations $T_{{\rho }_h},n\ge 2,nh=1$ of $T_{\rho }$ are obtained as follows:

$P_h^r\left(\rho u\right)={\int }_0^{1}r(y,x)\rho (y,x)u\left(x\right)dx$

$=h^{-1}{\int }_0^1{\sum }_{k=1,n}{I}_{h,k}\left(y\right)I_{h,k}\left(x\right)\rho (y,x)u\left(x\right)dx$

$={\int }_0^1\left[h^{-1}{\sum }_{k=1,n}{I}_{h,k}\left(y\right)\rho (y,x)I_{h,k}\left(x\right)\right]u\left(x\right)dx$

$={\int }_0^1{\rho }_h(y,x)u\left(x\right)dx$. So,

$T_{{\rho }_h}u:={\int }_0^1{\rho }_h(y,x)u\left(x\right)dx=P_h^r\left(\rho u\right)$.

Moreover, given the kernels $\phi$ and $\rho$, their approximations on $S_h$ verifies:

${\left(\phi \rho \right)}_h(y,x)=h^{-1}{\sum }_{k=1,n}{\left(\phi \rho \right)}_h^k(y,x)=h^{-1}{\sum }_{k=1,n}{\phi }_h^k(y,x){\sum }_{j=1,n}{\rho }_h^j(y,x)$

$=h^{-1}{\sum }_{k=1,n}{\phi }_h^k(y,x){\rho }_h^k(y,x):=h^{-1}{\sum }_{k=1,n}\left({\phi }_h^k{\rho }_h^k\right)(y,x)$ because the remaining terms are null from the disjoint supports.

Lemma 1. 

For any $T_{\rho }\in \mathcal{L}\left(H\right)$, the discretization schema (4-6) has the properties:

⋅the sequence of operator approximations$\{T_{{\rho }_h};n\ge 2,nh=1\}$is strongly convergent, that is the convergence property i) holds.

⋅The matrix representations of the operator approximations over the subspaces in F are one-diagonal sparse matrices.

⋅ If the diagonal entries of the matrix representations are strictly positive valued, $d_{kk}^h>0,\forall k=1,n,nh=1$, then the operator approximations $T_{{\rho }_h}$ are strictly positive i.e. the positivity property ii) holds.

Proof. 

From $T_{\rho }u-T_{{\rho }_h}u={\int }_0^1(\rho u-{\rho }_{h}u)dx={\int }_0^1(I-P_h^r)\left(\rho u\right)dx$. Then, $\parallel T_{\rho }u-T_{{\rho }_h}u\parallel \le \parallel I-P_h^r\parallel \parallel \rho u\parallel$.

Because $\{P_h^r;n\ge 2,nh=1\}$ is a sequence of orthogonal projections onto a family of including subspaces whose union is dense, $\parallel I-P_h^r\parallel \to 0$ with $n\to \infty ,nh=1$ holds; then the convergence of operator approximations property i) is satisfied:

${ϵ}_n:=\parallel T_{\rho }-T_{{\rho }_h}\parallel \to 0$ for $n\to 0$, nh=1.

Now, evaluating $T_{{\rho }_h}v$ for $v=I_{h,i}$, we obtain

$\left(T_{{\rho }_h}{I}_{h,i}\right)\left(y\right)=h^{-1}{\sum }_{k=1,n}{\int }_{{\Delta }_{h,k}}\left[I_{h,k}\left(y\right)\rho (y,x)I_{h,k}\left(x\right)\right]I_{h,i}\left(x\right)dx$

= $h^{-1}\left[{\int }_{{\Delta }_{h,i}}\rho (y,x)I_{h,i}\left(x\right)dx\right]I_{h,i}\left(y\right):=h^{-1}w\left(y\right)I_{h,i}\left(y\right)$

where $w=\left[{\int }_{{\Delta }_{h,i}}\rho (y,x)I_{h,i}\left(x\right)dx\right]\notin S_h$.

Because $\left(w{I}_{h,i}\right)$ has the support ${\Delta }_{h,i}$ (Note 2.c), $I_{h,j}$ has the support ${\Delta }_{h,j}$ and ${\Delta }_{h,i}\cap {\Delta }_{h,j}=\varnothing$ for $i\ne j$, we obtain

$\langle T_{{\rho }_h}{I}_{h,i},I_{h,j}\rangle =\langle w\left(y\right)I_{h,i}\left(y\right),I_{h,j}\left(y\right)\rangle =0$. Thus, the matrix representation of the finite rank operator $T_{{\rho }_h}$ on the basis $\{I_{h,k},k=1,n\}$ of $S_h$, is:

$M_h^r\left(T_{\rho }\right)=\mathrm{diag}{\left[h^{-1}{d}_{kk}^h\right]}_{k=1,n}$ where

$$d_{kk}^h={\int }_{{\Delta }_{h,k}}{\int }_{{\Delta }_{h,k}}{I}_{h,k}\left(y\right)\rho (y,x)I_{h,k}\left(x\right)dxdy:={\int }_{{\Delta }_{h,k}}{\int }_{{\Delta }_{h,k}}\rho (y,x)dxdy$$

(7)

and $d_{ij}^h=0$ for any pair (i,j), $i\ne j$ showing that the matrix is 1-diagonal.

Given $v_h={\sum }_{k=1,n}{c}_k{I}_{h,k}\in S_h$, and from $\parallel v_h{\parallel }^2=h{\sum }_{k=1,n}{c}_k\overline{c_k}$, we obtain:

$\langle T_{{\rho }_h}{v}_h,v_h\rangle =h^{-1}{\sum }_{k=1,n}{c}_k\overline{c_k}{d}_{kk}^h\ge {\alpha }_h\left(T_{{\rho }_h}\right){\parallel v_h\parallel }^2$

where ${\alpha }_h\left(T_{{\rho }_h}\right)$ is the positivity parameter of the operator approximation $T_{{\rho }_h}$,

$${\alpha }_h\left(T_{{\rho }_h}\right)=h^{-2}mi{n}_{(k=1,n)}{d}_{kk}^h>0$$

(8)

that is strict positive for any $S_h$ because the minimum from a finite number of strict positive quantities $d_{kk}^h$, $k=1,n,nh=1$ can not be zero.

□

Lemma 2. 

(Criterion for Positive valued kernels)). If the kernel of the integral operator $T_{\rho }\in \mathcal{L}\left(H\right)$ is strictly positive valued on (0,1)² except for a set of measure Lebesgue 0 and the operator approximations sequence of the positivity parameters verifies for some $\alpha >0$

iii) ${\alpha }_h\ge \alpha$for any$n\ge 2$, nh=1then$N_{T_{\rho }}=\left\{0\right\}$.

Proof. 

Let $\rho (y,x)\ge 0$ for any pair (x,y)$\in {(0,1)}^2$. Then, in (7) $d_{kk}^h>0$ because the set of the points (x,y) in the domain of $\rho$ are of measure Lebesgue zero, for any k=1,n, nh=1, therefore, ii) holds meaning we have the positivity of the operator approximations $T_{{\rho }_h}$, $n\ge 2$, nh=1. Then from iii), $N_{T_{\rho }}=\left\{0\right\}$.

□

Lemma 3. 

(Main Criterion)). If the operator approximations of $T_{\rho }\in \mathcal{L}\left(H\right)$ on schema (4-6) are positive and its sequence of positive parameters is bounded, then$N_{T_{\rho }}=\left\{0\right\}$.

Proof. 

The sequence $T_{{\rho }_h}$ converges using the approximation schema for $T_{\rho }$. Suppose that this sequence is also positive, that is $\langle T_{{\rho }_h}v,v\rangle \ge {\alpha }_h\left(T_{{\rho }_h}\right){\parallel v\parallel }^2>0$ for any $v\in S_h$.

Let the positivity parameters in (8) be bounded, that is $\alpha >0$ exists such that ${\alpha }_h\left(T_{{\rho }_h}\right)\ge \alpha$. Then, $E\cap N_{T_{\rho }}=\varnothing$. Because there are no zeros of $T_{\rho }$ in S, $N_{T_{\rho }}=\left\{0\right\}$.

□

The criterion defined by Lemma 2 as well the Corollary are versions of the Main Criterion, both covering the positivity (ii)) of the operator approximations. Lemma 3 (Main Criterion) is in fact Theorem 3 adapted to a discretization schema (4)-(6).

4. Alcantara-Bode’s Equivalence

The Hilbert-Schmidt integral operator $T_{\rho }$, with the kernel function $\rho (y,x)=\{y/x\}$ the fractional part of the quantity between brackets,

$\left(T_{\rho }u\right)\left(y\right)={\int }_0^1\rho (y,x)u\left(x\right)dx,u\in L^2(0,1)$

was used by Alcantara-Bode ([2]) for the equivalent formulation of the Riemann Hypothesis obtained from the Beurling equivalent formulation ([4]). Riemann conjectured (see [7]) that the Riemann Zeta function defined by the infinite sum:

$\zeta \left(s\right)=1+1/2^s+1/3^s+1/4^s+...$

has non trivial zeros s = $\sigma$ + it on the vertical line $\sigma =1/2$.

A connection between the Riemann Zeta function $\zeta$ and the integral operator $T_{\rho }$ can be found in [4] after reformulating as: $\left(T_{\rho }{x}^{s-1}\right)\left(\theta \right):={\int }_0^1\rho (\theta /x)x^{s-1}dx$, the left term in the equality below

${\int }_0^1\rho (\theta /x)x^{s-1}dx=\theta /(s-1)-{\theta }^s\zeta \left(s\right)/s,\sigma >0,s=\sigma +it$.

The kernel function $\rho \in L^2{(0,1)}^2$ is continue and positive valued almost everywhere, and the discontinuities in ${(0,1)}^2$ consists of a set of numerable one dimensional lines of the form $y=kx,k\in N$, being of Lebesgue measure zero and thus, Riemann integrable. The integral operator $T_{\rho }$ is a Hilbert-Schmidt operator ([2]), therefore $T_{\rho }\in \mathcal{L}\left(H\right)$ together with its associated Hermitian. Then, for it will consider an operator approximation sequence built on a multi-level structure by (4-6).

Because the operator approximations are convergent (Lemma 1), following the algorithm we should compute the diagonal entries in the matrix representation, $M_h^r\left(T_{\rho }\right)$ which are given by: $d_{kk}^h={\int }_{{\Delta }_{h,k}}{\int }_{{\Delta }_{h,k}}\rho (y,x)dxdy$, computed using the observation: for $x,y\in {\Delta }_{h,k}$$\{y/x\}=y/x$ for $y/x\le 1$ and $\{y/x\}=y/x-1$ for $y/x>1$, valued as:

$d_{11}^h=h^2(3-2\gamma )/4;d_{kk}^h={\displaystyle \frac{h^2}{2}}(-1+{\displaystyle \frac{2k-1}{k-1}}ln{\left({\displaystyle \frac{k}{k-1}}\right)}^{k-1}),\mathrm{for}k\ge 2$

where $\gamma$ is the Euler-Mascheroni constant. The sequence $\{f_k;k=2,n,nh=1\}$ defined by

$f_k:=h^{-2}{d}_{kk}^h=(-1+{\displaystyle \frac{2k-1}{k-1}}ln{\left({\displaystyle \frac{k}{k-1}}\right)}^{k-1})/2$ is decreasing for $k\ge 2$ and converges to 0.5 for k $\to \infty$. For $k\ge 3$, we have: $d_{22}^h>d_{kk}^h>0.5h^2>d_{11}^h$. Then from ${\alpha }_h\left(T_{{\rho }_h}\right)=h^{-2}{d}_{11}^h$ (8) we have:

${\alpha }_h\left(T_{{\rho }_h}\right)\ge \alpha :=(3-2\gamma )/4>0,\forall h,nh=1$.

Theorem 5. 

The Alcantara-Bode equivalent of RH holds, that is $N_{T_{\rho }}=\left\{0\right\}$.

Proof. 

Applying Lemma 2 (the kernel $\rho$ is positive valued a.e. on H) or Lemma 3 (the operator approximations are built by schema (4-6)), from ${\alpha }_h=\alpha =(3-2\gamma )/4>0$ a constant for any n, $nh=1$ we obtain $N_{T_{\rho }}=\left\{0\right\}$ proving that the Alcantara-Bode equivalent holds. □

Proof. 

The associated Hermitian integral operator has the kernel function $\widehat{\rho }(y,x):={\int }_0^1\rho (y,t)\overline{\rho (t,x)}dt={\int }_0^1\rho (y,t)\rho (t,x)dt={\int }_0^1\phi (y,x,t)dt$. We have to compute the corresponding entries in the matrix representations given by (7). For $y,x\in {\Delta }_{h,k},t\in {\Delta }_{h,j}$ and with $\lfloor \rfloor$ the floor function, the function $\phi$ to be integrated in variable t has the form

$\phi (y,x,t)=(y/t-\lfloor y/t\rfloor )(t/x-\lfloor t/x\rfloor )$

= $y/x-(t/x)\lfloor y/t\rfloor =(1/x)(y-t\lfloor y/t\rfloor )$ for $j<k,(t<kh)$:

= $y/x-(y/t)\lfloor t/x\rfloor =y(1/x-1/t\lfloor t/x\rfloor )$ for $j>k,(t>kh)$

Then with $d_{kk}^{h,j}:={\int }_{t\in {\Delta }_{h,j}}{\int }_{x\in {\Delta }_{h,k}}{\int }_{y\in {\Delta }_{h,k}}\phi (y,x,t)dtdxdy$,

${\widehat{d}}_{kk}^h={\sum }_{j=1,n}{d}_{kk}^{h,j}\ge n·mi{n}_{j=1,n}{d}_{kk}^{h,j}$. To compute ${\widehat{d}}_{kk}^h$, we will separate the terms in the sum as follows: $d_{kk}^{h,j=k}$, $d_{kk}^{h,j<k}$ and $d_{kk}^{h,j>k}$. The sequences $d_{kk}^{h,1<j<k}$ and $d_{kk}^{h,j>k}$ are increasing monotonically therefore, their components with the minimum values are $d_{kk}^{h,2}$ and $d_{kk}^{h,k}$. The term $d_{kk}^{h,1}\ge d_{kk}^{h,2}/8$. Therefore, for any $n\ge 2,nh=1$, the positivity parameter on $S_h$ is given by ${\alpha }_h\left(T_{{\widehat{\rho }}_h}\right)=h^{-2}mi{n}_{k=1,n}{\widehat{d}}_{kk}^h\ge {\displaystyle \frac{ln2}{16}}>0$. From Corollary, $N_{T_{\rho }}=\left\{0\right\}$.

□

Note 3. On the connection between discretization schema (4)-(6) and the restrictions schema (Note 1) on the dense set S built in par.3.

Let denote the restriction of $T_{\rho }$ on $S_h$ by ${T|}_{S_h}$. From positivity on $S_h$ mentioning that for $u\in E$ its orthogonal projection verifies $P_{h}u:=u_h=P_h{u}_h$, if $T_{\rho }u=0$ then:

${\alpha }_h\left(T{|}_{S_h}\right){\parallel u_h\parallel }^2\le \langle T{|}_{S_h}{u}_h,u_h\rangle =\langle P_{h}T{P}_h{u}_h,u_h\rangle$ = $\langle P_{h}T(u_h-u),u_h\rangle =\langle (u_h-u),P_h{T}^{*}{u}_h\rangle$

$\le {\beta }_h{\omega }_h\parallel u_h\parallel$,

a relationship for the restriction operator analogue with the one using the approximation operator $T_{{\rho }_h}$ and so, allowing the inequality between the parameters ${\alpha }_h,{\beta }_h,{\omega }_h,{\mu }_h$ for obtaining the desired relationship that infirm the existence of eligible zeros. If we observe that $\langle T{|}_{s_h}{u}_h,u_h\rangle =\langle P_{h}T{P}_h{u}_h,u_h\rangle$, meaning that the operator $\left(P_{h}T{P}_h\right)$ could be defined as $\left(P_h{T}_{\rho }{P}_h\right)=h{T}_{{\rho }_h}$. Then, ${\alpha }_h\left(T{|}_{S_h}\right)=h{\alpha }_h\left(T_{{\rho }_h}\right)$.

So, we point out that the solution given in [11], and presented here in Note 1 in a simplified form could be included here as part of the theory built later with [1] for dealing with operator restrictions instead of operator approximations.