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Invariant Subspaces for Spherical Contractions
Author(s) -
Eschmeier J
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000300
Subject(s) - mathematics , linear subspace , invariant subspace , invariant (physics) , pure mathematics , hilbert space , unit sphere , subspace topology , reflexive operator algebra , combinatorics , contraction (grammar) , mathematical analysis , mathematical physics , compact operator , medicine , extension (predicate logic) , computer science , programming language
Let T be a contraction on a complex Hilbert space H . A result of Brown, Chevreau and Pearcy from 1979 shows that T has a non‐trivial invariant subspace if the spectrum of T is dominating in the open unit disc. It is the purpose of the present paper to prove the multidimensional analogue of this result for spherical contractions T ∈ L ( H ) n that possess a spherical dilation and for which the Harte spectrum is dominating in the open unit ball B in C n . If even the essential Harte spectrum of T is dominating in B , then T is shown to be reflexive and to possess an extremely rich invariant subspace lattice. The proof is based on the existence of an H ∞ ‐functional calculus for completely non‐unitary spherical contractions and on a multidimensional analogue of the classical result of Sz.Nagy and Foias, stating that each spherical contraction which is neither of type C ·0 nor of type C 0· and which does not consist of multiples of the identity operator on H , possesses non‐trivial joint hyperinvariant subspaces. 1991 Mathematics Subject Classification : 47A13, 47A15, 47A60