On weak* continuous operators on ℬ(ℋ)

Weber, Robert E.

doi:10.1090/S0002-9939-1981-0630046-8

On weak$^{\ast }$ continuous operators on $\mathcal {B}(\mathcal {H})$
HTML articles powered by AMS MathViewer

by Robert E. Weber PDF

Proc. Amer. Math. Soc. 83 (1981), 735-742 Request permission

Abstract:

If $\Delta$ is a weak* continuous bounded linear operator on $\mathcal {B}(\mathcal {H})$ that fixes the ideal of compact operators $\mathcal {K}$ and ${\Delta _0}$ and $\delta$ are the induced maps on $\mathcal {K}$ and $\mathcal {B}(\mathcal {H})/\mathcal {K}$ then it is shown that $\Delta$ has closed range, has dense range, is bounded below, or is onto if and only if both ${\Delta _0}$ and $\delta$ have the same property. These results are then applied to the operator $X \to AXB$.

References

Chandler Davis and Peter Rosenthal, Solving linear operator equations, Canadian J. Math. 26 (1974), 1384–1389. MR 355649, DOI 10.4153/CJM-1974-132-6
J. Dixmier, Les fonctionnelles linéaires sur l’ensemble des opérateurs bornés d’un espace de Hilbert, Ann. of Math. (2) 51 (1950), 387–408 (French). MR 33445, DOI 10.2307/1969331
L. Fialkow, A note on the operator $X\rightarrow AX-XB$, Trans. Amer. Math. Soc. 243 (1978), 147–168. MR 502900, DOI 10.1090/S0002-9947-1978-0502900-3
Lawrence A. Fialkow, A note on norm ideals and the operator $X\longrightarrow AX-XB$, Israel J. Math. 32 (1979), no. 4, 331–348. MR 571087, DOI 10.1007/BF02760462
Lawrence A. Fialkow, A note on the range of the operator $X\rightarrow AX-XB$, Illinois J. Math. 25 (1981), no. 1, 112–124. MR 602902
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. (Szeged) 33 (1972), 179–192. MR 322534
Paul R. Halmos, Commutators of operators. II, Amer. J. Math. 76 (1954), 191–198. MR 59484, DOI 10.2307/2372409
B. E. Johnson and J. P. Williams, The range of a normal derivation, Pacific J. Math. 58 (1975), no. 1, 105–122. MR 380490, DOI 10.2140/pjm.1975.58.105
Gunter Lumer and Marvin Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41. MR 104167, DOI 10.1090/S0002-9939-1959-0104167-0
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
R. E. Weber, Analytic functions, ideals, and derivation ranges, Proc. Amer. Math. Soc. 40 (1973), 492–496. MR 353025, DOI 10.1090/S0002-9939-1973-0353025-2
J. P. Williams, On the range of a derivation, Pacific J. Math. 38 (1971), 273–279. MR 308809, DOI 10.2140/pjm.1971.38.273