Self-adjoint abstract differential operators

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self-adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one-dimensional Galilei, Poincaré, and Newton–Hooke groups as well as the two-dimensional Euclidean group.

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

Simplicity of linear ordinary differential operators

A continuum of unusual self-adjoint linear partial differential operators

Relative form boundedness and compactness for a second-order differential operator

Chapter 9 deals with trace class perturbations of parts of the spectrum of self-adjoint operators. In the first section, the decomposition of a self-adjoint operator and its spectrum into a pure point part, a singularly continuous part, and an absolutely continuous part is obtained. Let A be a self-adjoint operator, and D a self-adjoint trace class operator acting on the same Hilbert space, and set B=A+D. Krein’s spectral shift associated with such a pair {B,A} is a function on the real line which involves an integral representation formula for the trace of the difference of functions of A and B. The spectral shift and its properties and the proof of Krein’s trace formula are central themes of this chapter. We follow M.G. Krein’s original approach to the spectral shift which was based on perturbation determinants. A short introduction into perturbation determinants is given in a separate section. Another main topic of Chap. 9 is the spectral theory of rank one perturbations B=A+α〈⋅,u〉u of the operator A, where α is a real number, and u is a unit vector. We prove the Aronszajn–Donoghue theorem. It characterizes the various parts of the spectrum of the self-adjoint operator B in terms of boundary values on ℝ of the holomorphic function 〈R z (A)u,u〉.

The spectral representation of a symmetric nonmaximal operator transforms it into a multiplication operator by a complex variable acting in a linear space of pairs of vectorvalued functions holomorphic on the upper and lower half- planes. Each generalized spectral function of the same symmetric operator leads to its realization in form of a multiplication operator by a real variable in some Hilbert space. We study the connection between such realizations of the given operator and its spectral representation.

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

In this paper, we study the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for describing irreversible processes. The particular case of extensions parametrized by normal states is discussed in [1, 2]. The construction used below is similar to that of self-ajoint extensions for densely-defined, bounded below, symmetric operators in a Hilbert space. The dynamic semigroup, unlike the unitary operator group, acts not on elements of a Hilbert space 7/but on elements of a topological operator algebra in a Hilbert space B(H); moreover, this semigroup is completely positive (see [3]). For such semigroups, direct methods of describing domains of generating mappings are absent; this complicates the parametrization of extensions of the minimal generating mapping. It turns out that the range of the resolvent of the minimal dynamic semigroup belongs to the complement of the zero set (the kernel) of the limit mapping QOO(.) associated with the formal generating mapping of the dynamic semigroup. For this reason, we first consider general properties of the zero set for a completely positive mapping (see Sec. 1). In what follows, the C*-algebra generated by positive roots of a mapping Q(-) plays the central role. In Sec. 2, we prove that this algebra has the following property: it is maximal in the class of algebras that contain an element X if and only if X* X and X X* are elements of the algebra. In Sec. 3, the mentioned property is extended to the set of positive operators that are zeros of the limit mapping QOO(.) _- s-lira Qn (.). We prove that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal C*-Mgebra generated by positive roots of the mapping Q~(-) associated with the semigroup. In Sec. 4, we give a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Sec. 5, we prove the characteristic theorem for completely positive mappings with given ranges, which enables us to consider the set of conservative extensions as a *-weakly compact convex manifold in/3+(/2(7-~)), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements T in extreme points of E C T establishes a one-to-one correspondence between classes of equivalent normed measures on E and conservative extensions of the minimal dynamic semigroup; in this case, E plays the role of boundary and the class of equivalent measures plays the role of boundary condition for a conservative extension.

Let A 1 {A_1} and A 2 {A_2} be (unbounded) selfadjoint operators on a Hilbert space H \mathcal {H} which commute on a dense linear subspace of H \mathcal {H} . To conclude that A 1 {A_1} and A 2 {A_2} strongly commute, additional assumptions are necessary. Two propositions which contain such additional conditions are proved in §1. In §2 we define different commutants of unbounded operator algebras (form commutant, weak unbounded commutant, strong unbounded commutant) and we discuss the relations between them and their bounded parts. In §3 we construct a selfadjoint ∗ {*} -representation of the polynomial algebra in two variables for which the form commutant is different from the weak unbounded commutant.

Consider a self-adjoint operator H on a separable Hilbert space with non-trivial absolutely continuous component. We study the general properties of the real-valued functional, , which in quantum mechanics represents the sojourn time (or life time) of an initial state . We characterize the critical points of the sojourn time, τX, of the operator multiplication by x in , and prove that it attains a global maximum in the unit sphere of the Sobolev space .

We study the operator-norm error bound estimate for the exponential Trotter product formula in the case of accretive perturbations. LetA be a semibounded from below self-adjoint operator in a separable Hilbert space. LetB be a closed maximal accretive operator such that, together withB*, they are Kato-small with respect toA with relative bounds less than one. We show that in this case the operator-norm error bound estimate for the exponential Trotter product formula is the same as for the self-adjointB [12]: $$\parallel (e^{ - tA/n} e^{ - tB/n} )^n - e^{ - t(A + B)} \parallel \leqslant L\frac{{\ln n}}{n},n = 2,3,....$$