Spectral families of projections, semigroups, and differential operators

Benzinger, Harold E.; Berkson, Earl; Gillespie, T. A.

doi:10.1090/s0002-9947-1983-0682713-4

Cited by 27 publications

(17 citation statements)

References 18 publications

(5 reference statements)

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“…Nagy operators" (introduced in [20] and used in [1]) for the case of parabolic isometric groups in HP(D), 1 < p ::; 2 (Corollary (3.10)); in addition, the inner-outer factorization of (e(A)J)(Z) for J(z) E Y, in the parabolic case and 1 < p < +00, will be examined (Corollary (3.6)) and other related features that explore the role the above-mentioned spectral projections play in the structure of H P (D). Finally, in §4, we will obtain the corresponding results, mainly for the parabolic isometric group, on X = H P (T 2 ) , 1 < p < +00, using the representation of Berkson and Porta [8] and by introducing a suitable, for our purposes, direct sum decomposition of H P (T2) (Lemma (4.5)).…”

Section: P(x))mentioning

confidence: 99%

“…We have the following results concerning rp/(z) [6,13]: (1) In the case p = 2 we examine only the one-parameter group of isometries on H2(D) whose action is described in Proposition (2.2).…”

Section: Preliminariesmentioning

confidence: 99%

“…In [1], H. Benzinger, E. Berkson, and T. A. Gillespie, adopting an abstract operator-theoretic approach that avoids vector measures, achieved a generalization to arbitrary Banach spaces of classical Stone's theorem for one-parameter unitary groups.…”

Section: Introductionmentioning

confidence: 99%

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Spectral Decompositions of One-Parameter Groups of Isometries on Hardy Spaces

Karayannakis¹

1989

Transactions of the American Mathematical Society

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ABSTRACT, Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on Hardy spaces of the disk D and the torus T2 are examined; a concrete description of the (pointwise) action of these decompositions is presented, mainly in the parabolic case, leading to a complete description of the action of the partial sum-operators of M, Riesz when carried from LP(R) to HP(D), I < p :::; 2. The (pointwise) action of the spectral decompositions of these isometric groups on HP(T 2 ), I < p < +00 , is also examined and concrete descriptions are derived, mainly in the parabolic case.

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Section: P(x))mentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Spectral Decompositions of One-Parameter Groups of Isometries on Hardy Spaces

Karayannakis¹

1989

Transactions of the American Mathematical Society

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“…(See [3], [11,Theorem 4.2.3] or the closely related [2,Theorem 5.37].) This sum converges in the norm of BðXÞ, but possibly only conditionally.…”

mentioning

confidence: 99%

Compact well-bounded operators

Qingping¹,

Doust²

2001

Glasgow Math. J.

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Abstract. Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.1991 Mathematics Subject Classification. Primary 47B40. Secondary 34L10, 47A60, 47B07.1. Introduction. Well-bounded operators are defined as those which possess a functional calculus for the absolutely continuous functions on some compact interval ½a; b of the real line. Well-bounded operators were introduced by Smart [17] and Ringrose [14] in order to provide a theory for Banach space operators that was similar to the successful theory of self-adjoint operators on Hilbert space, but which included operators whose spectral expansions may only converge conditionally.On a general Banach space the integral representation theorems which one obtains for these operators are much less satisfactory than those obtained for selfadjoint operators. Nonetheless, even on an arbitrary Banach space, every compact well-bounded operator can be written in the form

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“…A slightly stronger notion available is that of well-bounded operator of type (B). Such operators have been intensively studied in various settings by Berkson, Gillespie and others; see [1], [3], [4], [5] and [9], for example.…”

mentioning

confidence: 99%

Well-bounded operators of type (B) in a class of Banach spaces

Ricker

1987

J Aust Math Soc A

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It is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of Q,-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature. Various notions of self-adjointness have been developed for operators acting in Banach spaces, each capturing in some respect the basic properties of self-adjointness in Hilbert space. One such notion is that of well-boundedness, a concept introduced by Smart, [17], and first studied by Smart and Ringrose,[15,16,17]; see [8] for a comprehensive treatment of such operators. Well-bounded operators are associated with certain monotone, projection-valued functions on the real line R (not necessarily given by a spectral measure) and have available a form of spectral diagonalization similar to, but in general weaker than, that for selfadjoint operators in Hilbert space, which make them suitable for treating conditional convergence. A slightly stronger notion available is that of well-bounded operator of type (B). Such operators have been intensively studied in various settings by Berkson, Gillespie and others; see Whereas there is a close analogy in certain respect between scalar-type spectral operators with real spectrum, in the sense of Dunford, and well-bounded operators of type (B), there are also fundamental differences, even in the Hilbert space

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Spectral families of projections, semigroups, and differential operators

Cited by 27 publications

References 18 publications

Spectral Decompositions of One-Parameter Groups of Isometries on Hardy Spaces

Spectral Decompositions of One-Parameter Groups of Isometries on Hardy Spaces

Compact well-bounded operators

Well-bounded operators of type (B) in a class of Banach spaces

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