Matrix representations of multidimensional integral and ergodic operators
Introduction
Let be some domain. Consider the Hilbert space 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:In other words, is generated by: (1) multiplication operators A · (the dot · denotes the place of the operator argument ), where are bounded measurable M × M matrix-valued functions with complex entries defined on Ω; (2) integral operators where I is the one-dimensional set (line) defined byso, the operator integrates u in one coordinate xn producing the function that is constant along this coordinate; (3) operatorsare measurable mappings. In particular, the algebra contains various integral operators of Fredholm type, ergodic operators (which are based on Tm), 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 developed in, e.g., [21], [22], [23].
The main problems for operators from 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 is very complex. We try to find some discrete analogue of for which the functional calculus can be constructed explicitly. One of the most important requirements to is to be finite dimensional. Because in this case can be expressed in terms of matrix algebras for which the functional calculus is well known. If 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 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)where are some vertices such that Ωi are disjoint. Recall that is the Hilbert space of vector-valued functions acting on Ω, square-integrable with respect to the Lebesgue measure μ. For any and introduce the following elementary operators defined bywhere 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 is constant along xn in Ωi for n ∈ α. In fact, the operator translates the values of u(x) from the cube Ωj to the cube Ωi, then, it takes the average in Ωi along xn, n ∈ α and multiplies the average by A, and, finally, it puts zero values inside other cubes Ωr, r ≠ i. The examples of action of different operators is demonstrated in Fig. 1.
The operators provide the interaction between the components of u in the various micro-domains Ωi. Roughly speaking, if h > 0 is sufficiently small then Ω can be approximately represented by (4) and the discrete analogue of can be chosen asIn other words, is generated by for all and . Let us discuss why is a discrete analogue of defined by (1). The discrete analogues of the multiplication operators A · are the multiplication operators with piecewise constant functions i.e.The corresponding operators are expressed in terms of as follows:The integral operators, see (1) and (2), are also expressed in terms of :where I is the identity matrix,and a(m) is the mth entry of the vector a. The discrete analogues of change-of-variables operators Tm: Ω → Ω (see (3)) should be based on discrete mappings since . Taking arbitrary pm we constructThen the discrete analogues of (3) are expressed in terms of as follows:and δ is the Kronecker delta.
The above arguments show that can be considered as the discrete approximation of . As it is shown in Theorem 1.3 and Example 2 below, if h → 0 then the approximation becomes better and better. Moreover, is a closed C*-subalgebra of and, hence, deserves its own study. The properties of are enough for most practical applications.
Denoting by * the Hermitian conjugation, one can easily check the fundamental relationsHence, are basis elements and any operator has the formwhere are some M × M matrices. In practice, form (14) is available after taking an approximation of the initial operator . The question is how to find explicitly the spectrum of inverse square root etc. As mentioned above, if we provide a representation of in terms of simple matrix algebras, then the answers on all these questions become explicit. We denote the simple matrix algebras as n ⩾ 1. Introduce the following matricesand the following mappingwhere is given by (14). The next theorem is our main result. Theorem 1.1 The mapping π is the C*-isomorphism between C*-algebras and . The inverse mapping has the formwhere are the blocks of the matrix given by
Note that while the most of operators from are infinite-dimensional and even non-compact, the algebra has a finite dimension. We immediately obtain the following Corollary 1.2 (i) The operator is invertible if and only if all the matrices Bα are invertible. In this case, can be computed explicitly (ii) Generalizing (i) we can take rational functions f and writeThe extension to algebraic and transcendent functions f is also obvious. (iii) The trace and the determinant of can be defined asthey satisfy the usual propertieswhere is the identity operator, I is MS × MS identity matrix, and . (iv) Since π is the C*-isomorphism, the operator norm of can be computed explicitlywhere, recall that . Remark The algebrawith piecewise constant functions A acting on is considered in [24]. The difference between and is the presence of more general class of sets Ω and the addition of change-of-variables operators . In the case with the uniform partition of each of the segments [0,1) on p intervals (i.e. ), becomes a C*-subalgebra of :where are binomial coefficients. It is interesting to note that the generalization of the structure of simplifies the proof of main results.
Let us discuss a norm of approximation of continuous operators from by discrete operators from . For simplicity, consider the case and . The generalization to M > 1 is similar. Consider a multi-kernel operator of the form, common in applications,Here, we also use the notationNote that where is the complement to the set α. Consider the uniform partition of Ω onto pN identical cubeswhereis the representation of in the base p numeral system. Let us take the approximation of bywhereThe next proposition shows us that in the operator norm. Theorem 1.3 Suppose that Aα(k, xα) in (25) are real functions with bounded first derivatives. Then
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 is self-adjoint, i.e. . Then is self-adjoint andwhere Bδ means δ-neighbourhood and we can set . If then is invertible and there is p such that and and the following estimate is trueMoreover, 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.
Section snippets
Examples
Example 1 Consider the case and the classical Fredholm integral operators (see [18])with S-step (piecewise constant) kernelsSuch operators form an algebra isomorphic to (see, e.g., [19]). But, this algebra does not contain the identity operator (). Let us supplement it by adding new operatorsIn other words
Proof of Theorems 1.1 and 1.3
At first, let us consider the semigroup of subsetsand the corresponding C*-algebraThe identity element in this algebra is where ∅ is the empty set. All the basis elements are self-adjoint. Define the mapping Lemma 3.1 The mapping π1 is the C*-isomorphism. The inverse mapping is defined by Proof Consider the following basis in
Conclusion
We have shown that the analysis of mixed multidimensional integral and some type of ergodic operators can be explicitly reduced to the analysis of special matrices. This allows us to compute functions of such operators and their spectra explicitly with an arbitrary precision.
Funding
This paper is a contribution to the project M3 of the Collaborative Research Centre TRR 181 “Energy Transfer in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 274762653. This work is also supported by the RFBR (RFFI) grant No. 19-01-00094.
References (26)
- et al.
Effective masses for Laplacians on periodic graphs
J. Math. Anal. Appl.
(2016) A new symbolic method for solving linear two-point boundary value problems on the level of operators
J. Symb. Comput.
(2005)The algebra of integro-differential operators on an affine line and its modules
J. Pure Appl. Algebra
(2013)- et al.
On integro-differential algebras
J. Pure Appl. Algebra
(2014) Algebra of 2D periodic operators with local and perpendicular defects
J. Math. Anal. Appl.
(2016)- et al.
Band gap Green’s functions and localized oscillations
Proc. R. Soc. A
(2007) - et al.
Waves in lattices with imperfect junctions and localised defect modes
Proc. R. Soc. A
(2013) - et al.
Localised point defect states in asymptotic models of discrete lattices
Q. J. Mech. Appl. Math.
(2013) Algebra of multidimensional periodic operators with defects
J. Math. Anal. Appl.
(2015)Recovery of defects from the information at detectors
Inverse Probl.
(2016)