On commuting unbounded self-adjoint operators. II
type: Item
instance of: scholarly article
title: On commuting unbounded self-adjoint operators. II
zbMATH Open document ID: 0576.47015
DOI: 10.1007/BF01195869
author: Konrad Schmüdgen
author: Jürgen Friedrich
published in: Integral Equations and Operator Theory
publication date: 1984-01-01T00:00:00Z
review text: The paper deals with couples of (unbounded) self-adjoint operators A and B of a Hilbert space which commute on a common core for A and B. As begun in part I (see the preceding review), these couples are studied via commutators of bounded operators (i.e., the resolvents of A and B). The main objects are couples \(\{\) A,B\(\}\) of the classes \({\mathfrak N}_{rs}\) which are defined as follows. Let \(r,s\in {\mathbb{N}}\cup \{+\infty \}\). A couple \(\{\) A,B\(\}\) of self-adjoint operators A and B is said to be in \({\mathfrak N}_{rs}\) if there exists a linear subspace \({\mathcal D}\) of the Hilbert space such that \({\mathcal D}\subseteq {\mathcal D}(A^ jB^ k)\cap {\mathcal D}(B^ kA^ j)\) and \(A^ jB^ k\phi = B^ kA^ j\phi\) for all \(\phi\in {\mathcal D}\) and j,k\(\in {\mathbb{N}}\), \(j\leq r\), \(k\leq s\) and such that \({\mathcal D}\) is a core for A and B. The classes \({\mathfrak N}_{rs}\) are characterized in terms of commutators of the resolvents of A and B. The examples from part I are considered again in this more general context. It is shown (for instance) that the classes \({\mathfrak N}_{rs}\) are different, i.e., \({\mathfrak N}_{rs}\neq {\mathfrak N}_{r's'}\) if (r,s)\(\neq (r',s')\) and \(r,s,r',s'\in {\mathbb{N}}\cup \{+\infty \}\).
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 3921311
zbMATH Keywords: commuting self-adjoint operators
zbMATH Keywords: hyponormal
zbMATH Keywords: operators
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: vector-valued shift operator
zbMATH Keywords: commute on a common core
zbMATH Keywords: commutators of bounded operators
zbMATH Keywords: commutators of the resolvents
MaRDI profile type: Publication
cites work: Q4057181
cites work: Mosaics, principal functions, and mean motion in von Neumann algebras
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cites work: Conditions for two self-adjoint operators to commute or to satisfy the Weyl relation.
cites work: Analytic vectors
cites work: Commutators and systems of singular integral equations. I
cites work: Q5516366
cites work: Q3697644
cites work: On commuting unbounded self-adjoint operators. III
full work available at URL: https://doi.org/10.1007/bf01195869
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description: scientific article; zbMATH DE number 3921311