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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the operator equation $ AX+XB=Q$

Author: Jerome A. Goldstein
Journal: Proc. Amer. Math. Soc. 70 (1978), 31-34
MSC: Primary 47A60; Secondary 47B25
MathSciNet review: 0477836
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Abstract: Consider the operator equation $ (\ast)AX + XB = Q$; here A and B are (possibly unbounded) selfadjoint operators and Q is a bounded operator on a Hilbert space. The theory of one parameter semigroups of operators is applied to give a quick derivation of M. Rosenblum's formula for approximate solutions of $ (\ast)$. Sufficient conditions are given in order that $ (\ast)$ has a solution in the Schatten-von Neumann class $ {\mathcal{C}_p}$ if Q is in $ {\mathcal{C}_p}$. Finally a sufficient condition for solvability of $ (\ast)$ is given in terms of T. Kato's notion of smoothness.

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Keywords: Hilbert space, selfadjoint operator, operator equation, compact operator, Schatten-von Neumann class, one-parameter group of isometries
Article copyright: © Copyright 1978 American Mathematical Society