On normal operator logarithms

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

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)

Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space

In this chapter we shall prove the spectral theorem for self-adjoint (unbounded) operators A in Hilbert spaces. It says that for each A there exists a unique spectral measure E on a(A), so that A = ∫ t dE. In the sequel H is a complex Hilbert space.

In the traditional Hilbert space quantum mechanics, (bounded or unbounded) selfadjoint operators are taken as physical observables. In the present work a more comprehensive approach is used: observables are represented as normalised positive operator measures. We develop the spectral representation theory of generally unbounded selfadjoint operators in this more general setting of positive operator measures, paving the way for important physical applications in the sequel.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

Results of Schoenberg and others on limits of periodic splines as their order,m, goes to infinity are extended to sequences of D m -splines determined by the powers of an unbounded non-negative self-adjoint operator D on a Hilbert space,H, and an evaluation mapL fromH toR n . All such limits lie in the lowest frequencyn-dimensional invariant subspace for D,T* n . When each term in the sequence is the D m -spline whose image underL matches a fixed vector,y, (anL-interpolant), then the limit is theL-interpolant toy fromT* n . When the terms aresmoothing splines derived fromy then the limit exists when the smoothing parameter goes to 0 ast −m . Ift is not an eigenvalue,α i, of D, the limit is theL-least squares best fit toy fromT* l ,l=card {j: α j<t}.

In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\)). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

This survey is based on the works of Kiev mathematicians and others in the structure theory of pairs of bounded and unbounded self-adjoint operators that satisfy an algebraic relation.

We obtain some necessary or sufficient conditions for an operator on a complex separable Hilbert space to be expressible as a product of two normal operators. For example, it is shown that if T is such a product, then $$\dim \ker T \ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*)$$ . On the other hand, any operator T satisfying $$\dim \ker T^* \ge \dim \ker T$$ and $$\dim \ (\ker T\ \cap \ \overline{\mathrm {ran}\, T})\ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*$$ ) is a product of two normal operators. Such results complement our previous ones on the products of finitely many normal operators. We also obtain characterizations for products of two essentially normal operators.

We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, g -EP and g -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.

The main result is that if $\{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.

The rigged hilbert spaces approach in singular perturbation theory

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form \({-Au \in \partial\varphi(u)}\) as well ass dissipative initial value evolutions of the form \({-\dot{u}(t) - Au(t) + \omega{u}(t) \in \partial\varphi(l,u(l))}\) where \({\varphi}\) is a convex potential on an infinite dimensional space, A is a linear operator and \({\omega}\) is any scalar. The framework developed in the above mentioned paper reformulates these problems as \({0\in \bar{\partial}L(u)}\) and \({\dot{u}(t) \in \bar{\partial}L(t,u(t))}\) respectively, where \({\bar{\partial}L}\) is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term.