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A Double Commutant Theorem for Conjugate Selfadjoint Operators
| Content Provider | Semantic Scholar |
|---|---|
| Author | Möller, J. |
| Copyright Year | 2010 |
| Abstract | Let A be a bounded linear transformation on the complex separable Hilbert space H. If there is a conjugation Q on H such that A = QA*Q, we say that A is conjugate selfadjoint. In this note we examine commutativity properties of conjugate selfadjoint operators which possess cyclic vectors. 1. Preliminaries. Let 77 be a complex Hilbert space with a countably infinite basis, and let (/ g) denote the inner product of two vectors in 77. By 77 © 77 we mean the Hilbert space of vectors/ © g having inner product (fl®gvf2®g2) = (Uf2) + (g»g2)A linear manifold is a subset which is closed under vector addition and under multiplication by complex numbers. A subspace is a linear manifold which is closed in the norm topology induced by the inner product. The smallest subspace containing the set U T^oifn) TMm De denoted by V{/n}I1 Fls a subset of Hilbert space, clos F will denote the closure of F in the norm topology and F1" = {g|(g, f) = 0, / G F}. Whenever A is a continuous linear transformation on 77, its graph T(A) = {/ © Af\f G 77} is a subspace of 77 © 77. One can easily verify that r(.4)x = {A*(-f) ©/|/ G 77}, where A * is the adjoint of A. The germinal idea of representing a linear transformation through its associated graph subspace originated in the work of J. von Neumann [1]. Hereafter we shall refer to a continuous (or bounded) linear transformation as an operator. If A and B are operators on 77, we define (A © B)(f ®g) = Af@ Bg. The set of all operators on 77 that commute with A is called the commutant of A. This algebra will be denoted by {A}'. The double commutant of A, designated {A)", is the algebra of all operators on 77 that commute with every member of {A }'. It is self evident that {A }' is abelian if and only if {A}' = {A }". A transformation Q on 77 is said to be a conjugation if Q2 = I and (Qf, Qg) = (g,f) for every/and g in 77. Intuitively speaking, Q replaces each element of 77 by its conjugate with respect to the "real" subspace consisting of all fixed points of Q [3, p. 357]. If there is a conjugation Q such that A = QA *Q, we shall call A conjugate selfadjoint. The well-known Hankel operators [4] belong to this class. Received by the editors July 18, 1980 and, in revised form, January 12, 1981. 1980 Mathematics Subject Classification Primary 47A05; Secondary 46J99. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1981-083-03/S0002-9939-1981-0627679-1/S0002-9939-1981-0627679-1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
