Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] E. Berkson, R. J. Fleming, J. A. Goldstein, and J. Jamison, One-parameter groups of isometries on $ {\mathcal{C}_p}$ (to appear).
  • [2] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R. I., 1957. MR 0089373 (19:664d)
  • [3] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [4] -, Wave operators and similarly for some non-selfadjoint operators, Math. Ann. 162 (1966), 258-279. MR 0190801 (32:8211)
  • [5] G. K. Pedersen, On the operator equation $ HT + TH = 2K$, Indiana U. Math. J. 25 (1976), 1029-1033. MR 0470721 (57:10467)
  • [6] J. R. Ringrose, Compact non-self-adjoint operators, Van Nostrand, London, 1971.
  • [7] M. Rosenblum, The operator equation $ BX = XA = Q$ with self-adjoint A and B, Proc. Amer. Math. Soc. 20 (1969), 115-120. MR 0233214 (38:1537)
  • [8] R. Schatten, Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1960. MR 0119112 (22:9878)
  • [9] B. Sz.-Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. Szeged 11 (1974), 152-157. MR 0022309 (9:191b)
  • [10] K. Yosida, Functional analysis, Springer-Verlag, New York, 1965.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A60, 47B25

Retrieve articles in all journals with MSC: 47A60, 47B25

Additional Information

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

American Mathematical Society