Spectral subspaces of subscalar and related operators

Authors:
T. L. Miller, V. G. Miller and M. M. Neumann

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1483-1493

MSC (2000):
Primary 47A11; Secondary 47B37, 47B40

DOI:
https://doi.org/10.1090/S0002-9939-03-07217-4

Published electronically:
October 3, 2003

MathSciNet review:
2053356

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Abstract | References | Similar Articles | Additional Information

Abstract: For a bounded linear operator on a complex Banach space and a closed subset of the complex plane this note deals with algebraic representations of the corresponding analytic spectral subspace from local spectral theory. If is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown that for all sufficiently large integers where denotes the largest linear subspace of for which for all Moreover, for a wide class of operators that satisfy growth conditions of polynomial or Beurling type, it is shown that is closed and equal to

**1.**T. Bermúdez and M. González,*On the boundedness of the local resolvent function,*Integral Equations Operator Theory**34**(1999), 1-8. MR**2000b:47010****2.**K. Clancey,*Seminormal Operators,*Lecture Notes in Math.**742**, Springer-Verlag, New York, 1979. MR**81c:47002****3.**I. Colojoara and C. Foias,*Theory of Generalized Spectral Operators,*Gordon and Breach, New York, 1968. MR**52:15085****4.**P. C. Curtis and M. M. Neumann,*Nonanalytic functional calculi and spectral maximal spaces,*Pacific J. Math.**137**(1989), 65-85. MR**90h:47059****5.**J. Eschmeier and M. Putinar,*Bishop's condition**and rich extensions of linear operators*, Indiana Univ. Math. J.**37**(1988), 325-348. MR**89k:47051****6.**J. Eschmeier and M. Putinar,*Spectral Decompositions and Analytic Sheaves,*Clarendon Press, Oxford, 1996. MR**98h:47002****7.**C. Foias and F.-H. Vasilescu,*Nonanalytic local functional calculus*, Czechoslovak Math. J.**24**(99) (1974), 270-283. MR**51:8885****8.**B. E. Johnson,*Continuity of linear operators commuting with continuous linear operators*, Trans. Amer. Math. Soc.**128**(1967), 88-102. MR**35:4748****9.**B. E. Johnson and A. M. Sinclair,*Continuity of linear operators commuting with continuous linear operators II*, Trans. Amer. Math. Soc.**146**(1969), 533-540. MR**40:4791****10.**K. B. Laursen and M. M. Neumann,*An Introduction to Local Spectral Theory,*Clarendon Press, Oxford, 2000. MR**2001k:47002****11.**T. L. Miller, V. G. Miller, and M. M. Neumann,*Growth conditions and decomposable extensions,*Contemp. Math.,**321**(2003), 197-205.**12.**M. M. Neumann,*Decomposable operators and generalized intertwining linear transformations,*Operator Theory: Advances and Applications**28**(1988), 209-222. MR**89f:47049****13.**V. Pták and P. Vrbová,*On the spectral function of a normal operator*, Czechoslovak Math. J.**23**(98) (1973), 615-616. MR**49:1196****14.**M. Putinar,*Hyponormal operators are subscalar,*J. Operator Theory**12**(1984), 385-395. MR**85h:47027****15.**C. R. Putnam,*Ranges of normal and subnormal operators*, Michigan Math. J.**18**(1971), 33-36. MR**43:2550****16.**A. M. Sinclair,*Automatic Continuity of Linear Operators,*London Math. Soc. Lecture Note Series**21**, Cambridge University Press, Cambridge, 1976. MR**58:7011****17.**P. Vrbová,*Structure of maximal spectral spaces of generalized scalar operators*, Czechoslovak Math. J.**23**(98) (1973), 493-496. MR**47:9338****18.**R. Whitley,*Fuglede's commutativity theorem and*Canad. Math. Bull.**33**(1990), 331-334. MR**92f:47017****19.**J.-K. Yoo,*Local spectral theory for operators on Banach spaces*, Far East J. Math. Sci., Special Volume (2001), Part III, 303-311. MR**2002m:47005**

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Additional Information

**T. L. Miller**

Affiliation:
Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762

Email:
miller@math.msstate.edu

**V. G. Miller**

Affiliation:
Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762

Email:
vivien@math.msstate.edu

**M. M. Neumann**

Affiliation:
Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762

Email:
neumann@math.msstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07217-4

Received by editor(s):
August 22, 2002

Received by editor(s) in revised form:
January 14, 2003

Published electronically:
October 3, 2003

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2003
American Mathematical Society