Spectral mapping theorems for subnormal operators

Dudziak, James J.

doi:10.1016/0022-1236(84)90083-1

Cited by 10 publications

(7 citation statements)

References 4 publications

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“…The details of the proofs of the remaining results in this section are essentially the same s the proofs of the corresponding results in [12] and will not be repeated here. Under the same hypothesis s Corollary 2.…”

Section: Theorem Suppose That K Is An M-spectral Set For the Opermentioning

confidence: 99%

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Von Neumann operators are reflexive.

Conway

Dudziak

1990

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Von Neumann operators are reflexive Dedicated to Heran S. Collins, the thesis advisor oft he first author and the mathematical grandfather ofthe second, on the occasion ofhis retirement from Louisiana State UniversityBy John B. Conway at Bloomington and James J. Dudziak*) at Lewisburg A compact subset K of the complex plane is a spectral set for an operator T on a Hubert space Jif if the spectrum of T, σ (T), is contained in K and for every rational function / with poles off K, \\f(T)\\^\\f\\ K = sup{\f(z)\:zeK}. The operator T is called a von Neumann operator if the spectrum of T is itself a spectral set for T. It has been known for some time that a von Neumann operator has a non-trivial invariant subspace [1]. In this paper it will be shown that every von Neumann operator is reflexive. That is, if A is any operator such that AJi^M for every subspace M of 3f that is invariant for Γ, then A belongs to the weakly closed operator algebra generated by T and the identity.For any operator T on J^9 let H^(T] be the closure of {p(T) : p is a polynomial} in the weak operator topology (WOT) of @(Μ ) and let P 00 (T) denote the closure of {p(T):p is a polynomial} in the natural weak* topology that &(3if) has s the dual space of the trace class operators on J^. In the process of proving the main result of this paper it will also be shown that for a von Neumann operator T, ^(T) = P°°(T) and if £:Ρ°°(Τ) -> c is a weak* continuous linear functional, then there are vectors g and h in tf such that L(X) = for all X in P 00 (T) (Corollary 3. 2).Actually the main accomplishment of this work is probably not the result proclaimed in the title, but rather the development of a functional calculus for von Neumann operators. Let Q denote the non-peak points of the function algebra R (σ (T)) and let /1 Q denote the restriction of planar Lebesgue measure to Q.

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Section: Theorem Suppose That K Is An M-spectral Set For the Opermentioning

confidence: 99%

“…In [12] a collection of compact sets are singled out and termed nice. The precise definition of a nice compact set is technical and the reader is referred to [12].…”

Section: § 3 Consequences Of Any P°°(/i) Functional Calculusmentioning

confidence: 99%

Von Neumann operators are reflexive.

Conway

Dudziak

1990

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Again let A be a T-invariant algebra on K, with set Q of nonpeak points, and let JLI be a measure on K. Following Sarason [17], Chaumat [4], and Dudziak [7], it is possible to obtain more precise information about the structure of S(jLi)nQ°, where Q° is the interior of Q with respect to the complex plane. Rather than give formal proofs, we state the results and refer to Dudziak [7] for proofs along the lines we have in mind.…”

Section: The Local Dirichlet Conditionmentioning

confidence: 99%

“…Rather than give formal proofs, we state the results and refer to Dudziak [7] for proofs along the lines we have in mind.…”

Section: The Local Dirichlet Conditionmentioning

confidence: 99%

Weak-star continuous homomorphisms and a decomposition of orthogonal measures

Cole¹,

Gamelin²

1985

Annales De L’institut Fourier

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“…and N is its minimal normal extension acting on 3?, then a functional calculus for S was studied in [11] and further extensions were explored in [14,15,18]. These results illustrate an intimate connection between subnormal operator theory and analytic function theory.…”

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confidence: 92%

Towards a functional calculus for subnormal tuples: the minimal normal extension

Conway¹

1991

Trans. Amer. Math. Soc.

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Abstract. In this paper the study of a functional calculus for subnormal ntuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.An «-tuple of commuting operators S = (Sx, ... , Sn) on a Hubert space %? is subnormal if there is an «-tuple N = (Nx, ... , N ) of commuting normal operators on a Hubert space X that contains %? such that for 1 < j < n, Nj£? ç ß? and S¡ = N\ß?. For any commuting «-tuple of operators S there is a notion of spectrum, the Taylor spectrum of S [30] (also see [12]), denoted by er(S). This spectrum is a nonempty compact subset of C". In this general theory it is possible to define (S) for any function analytic in a neighborhood of ct(S) . This functional calculus generalizes the usual Riesz functional calculus for a single operator. When S is assumed to be a subnormal «-tuple, however, a much richer functional calculus is possible. In this case 0(S) can be defined for functions that are weak* limits of analytic functions and 4>(S) becomes a subnormal operator. The central question in this development is "What are the properties of this operator ?" More generally, if S is an «-tuple and l, ... ,

(S) = (x (S), ... , (S)) is a commuting ^-tuple of operators. If S is subnormal, then (S) can be defined for 4> = ( ) consisting of functions that are weak* limits

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Spectral mapping theorems for subnormal operators

Cited by 10 publications

References 4 publications

Von Neumann operators are reflexive.

Von Neumann operators are reflexive.

Weak-star continuous homomorphisms and a decomposition of orthogonal measures

Towards a functional calculus for subnormal tuples: the minimal normal extension

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