On the operator equation $AX+XB=Q$
HTML articles powered by AMS MathViewer
- by Jerome A. Goldstein PDF
- Proc. Amer. Math. Soc. 70 (1978), 31-34 Request permission
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
-
E. Berkson, R. J. Fleming, J. A. Goldstein, and J. Jamison, One-parameter groups of isometries on ${\mathcal {C}_p}$ (to appear).
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/66), 258–279. MR 190801, DOI 10.1007/BF01360915
- Gert K. Pedersen, On the operator equation $HT+TH=2K$, Indiana Univ. Math. J. 25 (1976), no. 11, 1029–1033. MR 470721, DOI 10.1512/iumj.1976.25.25082 J. R. Ringrose, Compact non-self-adjoint operators, Van Nostrand, London, 1971.
- Marvin Rosenblum, The operator equation $BX-XA=Q$ with self-adjoint $A$ and $B$, Proc. Amer. Math. Soc. 20 (1969), 115–120. MR 233214, DOI 10.1090/S0002-9939-1969-0233214-7
- Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112, DOI 10.1007/978-3-642-87652-3
- Béla de Sz. Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Univ. Szeged. Sect. Sci. Math. 11 (1947), 152–157. MR 22309 K. Yosida, Functional analysis, Springer-Verlag, New York, 1965.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 70 (1978), 31-34
- MSC: Primary 47A60; Secondary 47B25
- DOI: https://doi.org/10.1090/S0002-9939-1978-0477836-2
- MathSciNet review: 0477836