Abstract
For operatorsA andB on a Hilbert space ℋ, let τ denote the operator on ℒ(ℋ) defined by τ(X)=AX−XB. Several equivalent conditions are given for τ to be surjective or bounded below. Analogues of these results are given for the restrictions of τ to norm ideals, and the norms of these restrictions are estimated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. R. Allan,On one-sided inverses in Banach algebras of holomorphic vector-valued functions, J. London Math. Soc.42 (1967), 463–470, MR35#5939.
G. R. Allan,Holomorphic vector-valued functions on a domain of holomorphy, J. London Math. Soc.42 (1967), 509–513, MR35#5940.
J. Anderson and J. G. Stampfli,Commutators and compressions, Israel J. Math.10 (1971), 433–441.
T. Ando,Distance to the set of scalars, preprint.
C. Apostol and L. Zsido,Ideals in W * -algebras and the function η of A. Brown and C. Pearcy, Rev. Roumaine Math. Pures Appl.18 (1973), 1151–1170.
C. Apostol and K. Clancey,Generalized inverses and spectral theory, Trans. Amer. Math. Soc.215 (1976), 293–300.
C. Apostol and K. Clancey,On generalized resolvents, Proc. Amer. Math. Soc.58 (1976), 163–168.
A. Brown and K. Pearcy,Spectra of tensor products of operators, Proc. Amer. Math. Soc.17 (1966), 162–169.
A. Brown and C. Pearcy,Operators on Hilbert Space, Springer-Verlag, to appear.
A. Brown and C. Pearcy,On the spectra of derivations on norm ideals, submitted to Acta Sci. Math. (Szeged).
C. Davis and P. Rosenthal,Solving linear operator equations, Canad. J. Math.26 (1974), 1384–1389.
J. Deddens, private communication.
M. R. Embry and M. Rosenblum,Spectra, tensor products, and linear operator equations, Pacific J. Math.53 (1974), 95–107.
L. A. Fialkow,A note on the operator X→AX−XB Trans. Amer. Math. Soc. 243 (1978), 147–168.
P. A. Fillmore, J. G. Stampfli, and J. P. Williams,On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math.33 (1972), 179–192, MR48#896.
C. K. Fong,Some aspects of derivations on ℬ(ℋ), Part I, Seminar Notes, University of Toronto, 1978, preprint.
I. C. Gohberg and M. G. Krein,Introduction to the Theory of Linear Nonselfadjoint Operators, Vol. 18, Translations of Mathematical Monographs, Amer. Math. Soc., 1969.
P R. Halmos,A Hilbert Space Problem Book, Van Nostrand, Princeton, 1967.
P. R. Halmos,Capacity in Banach algebras, Indiana Univ. Math. J.20 (1970/71), 855–863, MR42#3569.
R. E. Harte,Spectral mapping theorems on a tensor product, Bull. Amer. Math. Soc.79 (1973), 367–372.
D. A. Herrero, private communication.
G. Lumer and M. Rosenblum,Linear operator equations, Proc. Amer. Math. Soc.10 (1959), 32–41.
H. Radjavi and P. Rosenthal,Matrices for operators and generators of ℬ(ℋ), J. London Math. Soc.2 (1970), 557–560.
F. Riesz and B. Sz.-Nagy,Functional Analysis. Ungar, New York, 1955.
M. Rosenblum,On the operator equation BX−XA=Q, Duke Math. J.23 (1956), 263–269.
R. Schatten,Norm Ideals of Completely Continuous Operators, Ergebnisse der Math., Springer-Verlag, 1960.
J. G. Stampfli,Which weighted shifts are subnormal, Pacific J. Math.17 (1966), 367–379, MR33#1740.
J. G. Stampfli,The norm of a derivation, Pacific J. Math33 (1970), 737–747.
J. G. Stampfli,Derivations on ℬ(ℋ): The range. Illinois J. Math.17 (1973), 518–524.
Author information
Authors and Affiliations
Additional information
The author gratefully acknowledges support by a grant from the National Science Foundation.
Rights and permissions
About this article
Cite this article
Fialkow, L.A. A note on norm ideals and the operatorX→AX−XB . Israel J. Math. 32, 331–348 (1979). https://doi.org/10.1007/BF02760462
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02760462