Matrix representations of multidimensional integral and ergodic operators

Abstract

We provide a representation of the C*-algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. The method can be useful in various applications, since many discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: (1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, (2) we find a wave function and spectral estimates for 3D discrete Schrödinger equation with planar, guided, local potential defects, and point sources. The accuracy of approximation of continuous multi-kernel integral operators by the operators with piecewise constant kernels is also discussed.

Introduction

Let $\Omega \subset {\mathbb{R}}^N$ be some domain. Consider the Hilbert space $L^2:=L^2(\Omega \to {\mathbb{C}}^M,\mu )$ of vector-valued functions acting on Ω (μ is the Lebesgue measure). The goal of the paper is the study of a discrete analogue of the following algebra of operators acting on L²:

$${\mathcal{A}}_{\mathrm{c}}=\text{Alg}(\mathbf{A}·,\int ·d{x}_1,\dots ,\int ·d{x}_N,\mathcal{T}).$$

In other words, ${\mathcal{A}}_{\mathrm{c}}$ is generated by: (1) multiplication operators A ·  (the dot  ·  denotes the place of the operator argument $\mathbf{u}={\left(u_m\right)}_{m=1}^M\in L^2$), where $\mathbf{A}=\mathbf{A}(x_1,\dots ,x_N)$ are bounded measurable M × M matrix-valued functions with complex entries defined on Ω; (2) integral operators $\int ·d{x}_n={\int }_{\Omega \cap I}·d{x}_n,$ where I is the one-dimensional set (line) defined by

$$I=I(x_1,\dots ,x_{n-1},x_{n+1},\dots ,x_N)=\{(x_1,\dots ,x_{n-1},y_n,x_{n+1},\dots ,x_N):y_n\in \mathbb{R}\},$$

so, the operator $\int \mathbf{u}(x_1,\dots ,x_N)d{x}_n$ integrates u in one coordinate x_n producing the function that is constant along this coordinate; (3) operators

$$\mathcal{T}\mathbf{u}\left(\mathbf{x}\right)={\left(u_m\left({\mathbf{T}}_m\left(\mathbf{x}\right)\right)\right)}_{m=1}^M,\mathbf{x}\in \Omega ,\text{where}{\mathbf{T}}_m:\Omega \to \Omega$$

are measurable mappings. In particular, the algebra ${\mathcal{A}}_{\mathrm{c}}$ contains various integral operators of Fredholm type, ergodic operators (which are based on T_m), and their combinations. All such operators have different applications in mathematical physics, e.g., they describe a propagation of waves and other phenomena in complex structures with defects [1], [2], [3], [4], [5], [6], diffusion [7], [8], thermodynamic processes [9], [10], random Schrödinger operators and various operators on discrete graphs [11], [12], [13], electromagnetic scattering [14]. Some general aspects of the connection between integral and ergodic operators are discussed in [15], [16], [17]. Integral operators and some of their finite-dimensional approximations are discussed in [18], [19], [20]. There are also useful purely algebraic approaches for various integro-differential modules over $C^{\infty }\left(\mathbb{R}\right)$ developed in, e.g., [21], [22], [23].

The main problems for operators from ${\mathcal{A}}_{\mathrm{c}}$ are to find the spectrum, to find the inverse operators, square roots, or, more generally, to construct the functional calculus on this algebra. The difficulty is that ${\mathcal{A}}_{\mathrm{c}}$ is very complex. We try to find some discrete analogue $\mathcal{A}$ of ${\mathcal{A}}_{\mathrm{c}}$ for which the functional calculus can be constructed explicitly. One of the most important requirements to $\mathcal{A}$ is to be finite dimensional. Because in this case $\mathcal{A}$ can be expressed in terms of matrix algebras for which the functional calculus is well known. If $\mathcal{A}$ is finite dimensional then, due to the Stone–Weierstrass theorem, all the matrix-valued functions A should be piecewise constant, otherwise the subalgebra generated by A has an infinite dimension. This tells us how the operator algebra $\mathcal{A}$ should be arranged. It is natural to suppose that Ω is a union of a finite number of shifted copies of a cube $H={[0,h)}^N$ (h > 0)

$$\Omega =\bigcup _{i=1}^S{\Omega }_i,{\Omega }_i={\mathbf{a}}_i+H,$$

where ${\mathbf{a}}_i\in {\mathbb{R}}^N$ are some vertices such that Ω_i are disjoint. Recall that $L^2:=L^2(\Omega \to {\mathbb{C}}^M,\mu )=L^2{\left(\Omega \right)}^M$ is the Hilbert space of vector-valued functions acting on Ω, square-integrable with respect to the Lebesgue measure μ. For any $\mathbf{A}\in {\mathbb{C}}^{M\times M},$ $\alpha \subset \{1,\dots ,N\}$ and $i,j\in \{1,\dots ,S\}$ introduce the following elementary operators ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]:L^2\to L^2$ defined by

$${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]\mathbf{u}\left(\mathbf{x}\right)=\{\begin{array}{cc}{h}^{-\left|\alpha \right|}\mathbf{A}{\int }_{{\Omega }_i}\mathbf{u}(\mathbf{x}-{\mathbf{a}}_i+{\mathbf{a}}_j)d{\mathbf{x}}_{\alpha },\hfill & \mathbf{x}\in {\Omega }_i,\hfill \\ \\ 0,\hfill & \mathbf{x}\in \Omega \setminus {\Omega }_i,\hfill \end{array}$$

where $d{\mathbf{x}}_{\alpha }={\prod }_{n\in \alpha }d{x}_n$ and the number of elements in α is denoted by |α|. Note that α can be the empty set ∅, in this case there is no ∫ in (5). Also note that ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]\mathbf{u}\left(\mathbf{x}\right)$ is constant along x_n in Ω_i for n ∈ α. In fact, the operator ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]$ translates the values of u(x) from the cube Ω_j to the cube Ω_i, then, it takes the average in Ω_i along x_n, n ∈ α and multiplies the average by A, and, finally, it puts zero values inside other cubes Ω_r, r ≠ i. The examples of action of different operators $\mathcal{E}$ is demonstrated in Fig. 1.

The operators ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]$ provide the interaction between the components of u in the various micro-domains Ω_i. Roughly speaking, if h > 0 is sufficiently small then Ω can be approximately represented by (4) and the discrete analogue of ${\mathcal{A}}_{\mathrm{c}}$ can be chosen as

$$\mathcal{A}=\text{Alg}\left({\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]\right).$$

In other words, $\mathcal{A}$ is generated by ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]$ for all $\mathbf{A}\in {\mathbb{C}}^{M\times M},$ $\alpha \subset \{1,\dots ,N\}$ and $i,j\in \{1,\dots ,S\}$. Let us discuss why $\mathcal{A}$ is a discrete analogue of ${\mathcal{A}}_{\mathrm{c}}$ defined by (1). The discrete analogues of the multiplication operators A ·  are the multiplication operators with piecewise constant functions $\mathbf{A}=\mathbf{A}\left(\mathbf{x}\right),$ i.e.

$$\mathbf{A}\left(\mathbf{x}\right)={\mathbf{A}}_i=\mathrm{const},\mathbf{x}\in {\Omega }_i,i=1,\dots ,S.$$

The corresponding operators are expressed in terms of $\mathcal{E}$ as follows:

$$\mathbf{A}·=\sum _{i=1}^S{\mathcal{E}}_{ii}^{\varnothing }\left[{\mathbf{A}}_i\right].$$

The integral operators, see (1) and (2), are also expressed in terms of $\mathcal{E}$:

$$\int ·d{x}_n=\sum _{i=1}^S\sum _{j\in {\beta }_i}h{\mathcal{E}}_{ij}^{\left\{n\right\}}\left[\mathbf{I}\right],$$

where I is the identity matrix,

$${\beta }_i=\{j:{\mathbf{a}}_j\left(m\right)={\mathbf{a}}_i\left(m\right),\forall m\ne n\}$$

and a(m) is the mth entry of the vector a. The discrete analogues of change-of-variables operators T_m: Ω → Ω (see (3)) should be based on discrete mappings $p_m:\{1,\dots ,S\}\to \{1,\dots ,S\},$ since $\Omega ={\cup }_{i=1}^S{\Omega }_i$. Taking arbitrary p_m we construct

$${\mathbf{T}}_m\mathbf{x}=\mathbf{x}-{\mathbf{a}}_i+{\mathbf{a}}_{p_m\left(i\right)},\mathbf{x}\in {\Omega }_i,i=1,\dots ,S.$$

Then the discrete analogues of $\mathcal{T}$ (3) are expressed in terms of $\mathcal{E}$ as follows:

$$\mathcal{T}=\sum _{m=1}^M\sum _{i=1}^S{\mathcal{E}}_{i,p_m\left(i\right)}^{\varnothing }\left[{\mathbf{I}}_m\right],\text{where}{\mathbf{I}}_m={\left({\delta }_{im}{\delta }_{jm}\right)}_{i,j=1}^M\in {\mathbb{C}}^{M\times M}$$

and δ is the Kronecker delta.

The above arguments show that $\mathcal{A}$ can be considered as the discrete approximation of ${\mathcal{A}}_c$. As it is shown in Theorem 1.3 and Example 2 below, if h → 0 then the approximation becomes better and better. Moreover, $\mathcal{A}$ is a closed C*-subalgebra of ${\mathcal{A}}_{\mathrm{c}}$ and, hence, deserves its own study. The properties of $\mathcal{A}$ are enough for most practical applications.

Denoting by * the Hermitian conjugation, one can easily check the fundamental relations

$${\mathcal{E}}_{ij}^{\alpha }{\left[\mathbf{A}\right]}^{*}={\mathcal{E}}_{ji}^{\alpha }\left[{\mathbf{A}}^{*}\right],{\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]+{\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{B}\right]={\mathcal{E}}_{ij}^{\alpha }[\mathbf{A}+\mathbf{B}],{\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]{\mathcal{E}}_{kl}^{\beta }\left[\mathbf{B}\right]={\delta }_{jk}{\mathcal{E}}_{il}^{\alpha \cup \beta }\left[\mathbf{A}\mathbf{B}\right].$$

Hence, ${\mathcal{E}}_{ij}^{\alpha }\left[\mathbf{A}\right]$ are basis elements and any operator $\mathcal{A}\in \mathcal{A}$ has the form

$$\mathcal{A}=\sum _{\alpha \subset \{1,\dots ,N\}}\sum _{i,j=1}^S{\mathcal{E}}_{ij}^{\alpha }\left[{\mathbf{A}}_{ij}^{\alpha }\right],$$

where ${\mathbf{A}}_{ij}^{\alpha }$ are some M × M matrices. In practice, form (14) is available after taking an approximation of the initial operator ${\mathcal{A}}_{\mathrm{c}}\in {\mathcal{A}}_{\mathrm{c}}$. The question is how to find explicitly the spectrum of $\mathcal{A},$ inverse ${\mathcal{A}}^{-1},$ square root $\sqrt{\mathcal{A}},$ etc. As mentioned above, if we provide a representation of $\mathcal{A}$ in terms of simple matrix algebras, then the answers on all these questions become explicit. We denote the simple matrix algebras as ${\mathbb{C}}^{n\times n},$ n ⩾ 1. Introduce the following matrices

$${\mathbf{A}}_{\alpha }={\left({\mathbf{A}}_{ij}^{\alpha }\right)}_{i,j=1}^S\in {\mathbb{C}}^{MS\times MS},{\mathbf{B}}_{\alpha }=\sum _{\beta \subset \alpha }{\mathbf{A}}_{\beta }$$

and the following mapping

$$\pi :\mathcal{A}\to {\left({\mathbb{C}}^{MS\times MS}\right)}^{2^N},\pi \left(\mathcal{A}\right)={\left({\mathbf{B}}_{\alpha }\right)}_{\alpha \subset \{1,\dots ,N\}},$$

where $\mathcal{A}$ is given by (14). The next theorem is our main result.

Theorem 1.1

The mapping π is the C*-isomorphism between C*-algebras $\mathcal{A}$ and ${\left({\mathbb{C}}^{MS\times MS}\right)}^{2^N}$. The inverse mapping has the form

$${\pi }^{-1}\left({\left({\mathbf{B}}_{\alpha }\right)}_{\alpha \subset \{1,\dots ,N\}}\right)=\sum _{\alpha \subset \{1,\dots ,N\}}\sum _{i,j=1}^S{\mathcal{E}}_{ij}^{\alpha }\left[{\mathbf{A}}_{ij}^{\alpha }\right],$$

where ${\mathbf{A}}_{ij}^{\alpha }\in {\mathbb{C}}^{M\times M}$ are the blocks of the matrix ${\mathbf{A}}_{\alpha }={\left({\mathbf{A}}_{ij}^{\alpha }\right)}_{i,j=1}^S$ given by

$${\mathbf{A}}_{\alpha }=\sum _{\beta \subset \alpha }{(-1)}^{|\alpha \setminus \beta |}{\mathbf{B}}_{\beta }.$$

Note that while the most of operators from $\mathcal{A}$ are infinite-dimensional and even non-compact, the algebra $\mathcal{A}$ has a finite dimension. We immediately obtain the following

Corollary 1.2

(i) The operator $\mathcal{A}$ is invertible if and only if all the matrices B_α are invertible. In this case, ${\mathcal{A}}^{-1}$ can be computed explicitly

$${\mathcal{A}}^{-1}={\pi }^{-1}\left({\left({\mathbf{B}}_{\alpha }^{-1}\right)}_{\alpha \subset \{1,\dots ,N\}}\right).$$

(ii) Generalizing (i) we can take rational functions f and write

$$f\left(\mathcal{A}\right)={\pi }^{-1}\left({\left(f\left({\mathbf{B}}_{\alpha }\right)\right)}_{\alpha \subset \{1,\dots ,N\}}\right).$$

The extension to algebraic and transcendent functions f is also obvious.

(iii) The trace and the determinant of $\mathcal{A}$ can be defined as

$$\det \left(\mathcal{A}\right)=\prod _{\alpha \subset \{1,\dots ,N\}}\det {\mathbf{B}}_{\alpha },\text{tr}\left(\mathcal{A}\right)=\sum _{\alpha \subset \{1,\dots ,N\}}\text{tr}{\mathbf{B}}_{\alpha },$$

they satisfy the usual properties

$$\det \left(\mathcal{A}\mathcal{B}\right)=\det \left(\mathcal{A}\right)\det \left(\mathcal{B}\right),\text{tr}(a\mathcal{A}+b\mathcal{B})=a\text{tr}\left(\mathcal{A}\right)+b\text{tr}\left(\mathcal{B}\right),$$

$$\det \left(e^{\mathcal{A}}\right)=e^{\text{tr}\left(\mathcal{A}\right)},\text{sp}\left(\mathcal{A}\right)=\{\lambda :\det (\mathcal{A}-\lambda \mathcal{I})=0\},$$

where $\mathcal{I}={\pi }^{-1}\left({\left(\mathbf{I}\right)}_{\alpha \subset \{1,\dots ,N\}}\right)$ is the identity operator, I is MS × MS identity matrix, and $a,b\in \mathbb{C},$ $\mathcal{A},\mathcal{B}\in \mathcal{A}$.

(iv) Since π is the C*-isomorphism, the operator norm of $\mathcal{A}\in \mathcal{A}$ can be computed explicitly

$${\parallel \mathcal{A}\parallel }_{L^2\to L^2}=\max \{\lambda :\lambda ares-valuesof{\mathbf{B}}_{\alpha }\},$$

where, recall that $\pi \left(\mathcal{A}\right)={\left({\mathbf{B}}_{\alpha }\right)}_{\alpha \subset \{1,\dots ,N\}}$.

Remark

The algebra

$$\mathcal{B}=\text{Alg}(\mathbf{A}·,{\int }_0^1·d{x}_1,\dots ,{\int }_0^1·d{x}_n)$$

with piecewise constant functions A acting on $\Omega ={[0,1)}^N$ is considered in [24]. The difference between $\mathcal{A}$ and $\mathcal{B}$ is the presence of more general class of sets Ω and the addition of change-of-variables operators $\mathcal{T}$. In the case $\Omega ={[0,1)}^N$ with the uniform partition of each of the segments [0,1) on p intervals (i.e. $S=p^N$), $\mathcal{B}$ becomes a C*-subalgebra of $\mathcal{A}$:

$$\mathcal{B}\cong \prod _{n=0}^N{\left({\mathbb{C}}^{M{p}^n\times M{p}^n}\right)}^{\left(\genfrac{}{}{0pt}{}{N}{n}\right)p^{N-n}}\subset {\left({\mathbb{C}}^{M{p}^N\times M{p}^N}\right)}^{2^N}\cong \mathcal{A},$$

where $\left(\genfrac{}{}{0pt}{}{N}{n}\right)$ are binomial coefficients. It is interesting to note that the generalization of the structure of $\mathcal{A}$ simplifies the proof of main results.

Let us discuss a norm of approximation of continuous operators from ${\mathcal{A}}_{\mathrm{c}}$ by discrete operators from $\mathcal{A}$. For simplicity, consider the case $\Omega ={[0,1)}^N$ and $M=1$. The generalization to M > 1 is similar. Consider a multi-kernel operator ${\mathcal{A}}_c:L^2\left(\Omega \right)\to L^2\left(\Omega \right)$ of the form, common in applications,

$${\mathcal{A}}_{c}u\left(\mathbf{k}\right)=\sum _{\alpha \subset \{1,\dots ,N\}}{\int }_{{[0,1)}^{\left|\alpha \right|}}{A}_{\alpha }(\mathbf{k},{\mathbf{x}}_{\alpha })u({\mathbf{k}}_{\overline{\alpha }}+{\mathbf{x}}_{\alpha })d{\mathbf{x}}_{\alpha },\mathbf{k}\in \Omega ,u\in L^2\left(\Omega \right).$$

Here, we also use the notation

$${\mathbf{x}}_{\alpha }=\left({\tilde{x}}_n\right),{\tilde{x}}_n=\{\begin{array}{cc}{x}_n,\hfill & n\in \alpha ,\hfill \\ \\ 0,\hfill & n\notin \alpha .\hfill \end{array}$$

Note that $\mathbf{x}={\mathbf{x}}_{\alpha }+{\mathbf{x}}_{\overline{\alpha }},$ where $\overline{\alpha }=\{1,\dots ,N\}\setminus \alpha$ is the complement to the set α. Consider the uniform partition of Ω onto p^N identical cubes

$$\Omega =\bigcup _{i=1}^{p^N}{\Omega }_i,{\Omega }_i={\mathbf{a}}_i+{[0,1/p)}^N,$$

where

$${\mathbf{a}}_i=\frac{1}{p^N}{\left(b_j\right)}_{j=1}^N,i-1=\sum _{j=1}^N{p}^{j-1}{b}_j,b_j\in \{0,\dots ,p-1\}$$

is the representation of $i-1$ in the base p numeral system. Let us take the approximation of ${\mathcal{A}}_c$ by

$$\mathcal{A}=\sum _{\alpha \subset \{1,\dots ,N\}}\sum _{i=1}^{p^N}\sum _{j=1}^{p^N}\frac{1}{p^{\left|\alpha \right|}}{\mathcal{E}}_{ij}^{\alpha }\left[A_{ij}^{\alpha }\right],$$

where

$$A_{ij}^{\alpha }=\{\begin{array}{cc}{A}_{\alpha }({\mathbf{a}}_i+\mathit{\varepsilon },{({\mathbf{a}}_j+\mathit{\varepsilon })}_{\alpha }),\hfill & {\left({\mathbf{a}}_i\right)}_{\overline{\alpha }}={\left({\mathbf{a}}_j\right)}_{\overline{\alpha }},\hfill \\ \\ 0,\hfill & otherwise,\hfill \end{array}\text{and}\mathit{\varepsilon }={(1/\left(2p\right))}_{r=1}^N.$$

The next proposition shows us that $\mathcal{A}\to {\mathcal{A}}_c$ in the operator norm.

Theorem 1.3

Suppose that A_α(k, x_α) in (25) are real functions with bounded first derivatives. Then

$$\parallel {\mathcal{A}}_c{-\mathcal{A}\parallel }_{L^2\to L^2}⩽\frac{N}{2p}\sum _{\alpha \subset \{1,\dots ,N\}}{\parallel \nabla A_{\alpha }\parallel }_{L^{\infty }}\to 0forp\to \infty .$$

Along with Theorem 1.1 and Corollary 1.2, we can use Theorem 1.3 to determine the spectrum and the inverse operator.

Corollary 1.4

Under the assumptions of Theorem 1.3, we assume also that ${\mathcal{A}}_c$ is self-adjoint, i.e. $A_{\alpha }(\mathbf{k},{\mathbf{x}}_{\alpha })=A_{\alpha }({\mathbf{k}}_{\overline{\alpha }}+{\mathbf{x}}_{\alpha },{\mathbf{k}}_{\alpha })$. Then $\mathcal{A}$ is self-adjoint and

$$\text{sp}\left({\mathcal{A}}_c\right)\subset B_{\delta }\left(\text{sp}\left(\mathcal{A}\right)\right),\text{sp}\left(\mathcal{A}\right)\subset B_{\delta }\left(\text{sp}\left({\mathcal{A}}_c\right)\right),$$

where B_δ means δ-neighbourhood and we can set $\delta =\frac{N}{2p}{\sum }_{\alpha \subset \{1,\dots ,N\}}{\parallel \nabla A_{\alpha }\parallel }_{L^{\infty }}$. If $0\notin \text{sp}\left({\mathcal{A}}_c\right)$ then ${\mathcal{A}}_c$ is invertible and there is p such that $0\notin \text{sp}\left(\mathcal{A}\right)$ and $\parallel {\mathcal{A}}^{-1}{\parallel }_{L^2\to L^2}⩽{\delta }^{-1},$ and the following estimate is true

$$\parallel {\mathcal{A}}_c^{-1}-{\mathcal{A}}^{-1}{\parallel }_{L^2\to L^2}⩽\frac{\delta \parallel {\mathcal{A}}^{-1}{\parallel }_{L^2\to L^2}^2}{1-\delta \parallel {\mathcal{A}}^{-1}{\parallel }_{L^2\to L^2}}.$$

Moreover, RHS of (32) tends to 0 for p → ∞.

The rest of the paper is organized as follows. Section 2 contains two examples: (1) new formulas for the functions of 1D Fredholm integral operators with step kernels; (2) the application of the method for obtaining a solution (with arbitrary precision) and spectral estimates for 3D discrete Schrödinger equation with planar, guided, and local potential defects. A short proof of the main result based on the explicit representation of a semigroup algebra of subsets is given in Section 3. We conclude in Section 4.