General Wiener-Hopf operators and the numerical range of an operator

Pellegrini, Victor J.

doi:10.1090/S0002-9939-1973-0315493-1

General Wiener-Hopf operators and the numerical range of an operator
HTML articles powered by AMS MathViewer

by Victor J. Pellegrini

Proc. Amer. Math. Soc. 38 (1973), 141-146

DOI: https://doi.org/10.1090/S0002-9939-1973-0315493-1

PDF | Request permission

Abstract:

Let $H$ be a separable Hilbert space and $A$ a bounded operator on $H$. For a selfadjoint projection $P$ on $H$ we consider the general Wiener-Hopf operator ${T_P}(A) = P{A_{R(P)}}$ where $R(P)$ denotes the range of $P$. In this paper we study the relation between ${T_P}(A)$ and $W(A)$, the numerical range of $A$. In particular we characterize those operators $A$ such that ${T_P}(A)$ is invertible for every $P$.

References

Allen Devinatz and Marvin Shinbrot, General Wiener-Hopf operators, Trans. Amer. Math. Soc. 145 (1969), 467–494. MR 251573, DOI 10.1090/S0002-9947-1969-0251573-0
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, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Marvin Shinbrot, On the range of general Wiener-Hopf operators, J. Math. Mech. 18 (1968/1969), 587–601. MR 0250135
James P. Williams, Similarity and the numerical range, J. Math. Anal. Appl. 26 (1969), 307–314. MR 240664, DOI 10.1016/0022-247X(69)90154-1