Eine metrische Kennzeichnung der selbstadjungierten Operatoren

Vidav, Ivan

doi:10.1007/bf01186601

Cited by 77 publications

(72 citation statements)

References 2 publications

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“…This notion was shown to coincide with the one previously introduced by Vidav in [20], namely, that 11/ + itTW = 1 + o(0 as t -► 0, t real.…”

Section: By Ahmed Ramzy Sourourmentioning

confidence: 67%

Semigroups of Scalar Type Operators on Banach Spaces

Sourour¹

1974

Transactions of the American Mathematical Society

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ABSTRACT. The main result is that if {T(t): t > 0} is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space X and if the resolutions of the identity for T(t) are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure K( ■ ) such that T(t) = { exp (Kt)dK(K), for t > 0. This is a direct generalization of the well-known theorem of Sz. 1. Introduction. In this paper we study strongly continuous semigroups {71(f): t > 0} of scalar type operators on a Banach space and extend some wellknown results about semigroups of normal or selfadjoint operators on a Hilbert space. The main result (Theorem 5.3) generalizes Sz.-Nagy's theorem about semigroups of normal operator [12, Theorem 22.4.2] or [19, §XI 3]. We do not assume that the resolutions of the identity for the operators T(t) generate a bounded Boolean algebra of projections; this is obtained as a result. If one assumes this, a much simpler proof of the main theorem could be given. However, this would be quite restrictive since in general the resolutions of the identity for two commuting scalar type operators do not always generate a bounded Boolean algebra of projections, even on a reflexive space (see [16]).In §6 we extend a theorem of Foias, [11] on semigroups of scalar type operators on a Hilbert space. In §7 we study the continuity of f(T(t)), as a

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“…This notion was shown to coincide with the one previously introduced by Vidav in [20], namely, that 11/ + itTW = 1 + o(0 as t -► 0, t real.…”

Section: By Ahmed Ramzy Sourourmentioning

confidence: 67%

Semigroups of Scalar Type Operators on Banach Spaces

Sourour¹

1974

Transactions of the American Mathematical Society

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“…In view of Theorem 1 of Vidav [9], the hypothesis of the lemma and the fact that and the fact that B is commutative, the lemma follows as soon as we establish that S = U + iV for some U and V in B with U, V hermitian in || ||. But this can be shown by an argument similar to that given by Berkson in Theorem 3.1 in [1].…”

Section: Proof Let D = I \J {R T J T ; T E A}mentioning

confidence: 87%

“…Since by (1.3), R -R λ is hermitian in some equivalent norm, by a result of Vidav [9], it follows that R -R x = 0. Hence R = R x .…”

Section: Theorem 1 If S Is a Scalar Type Operator On A Banach Space mentioning

confidence: 99%

“…Further as a consequence of Lemma 1 of Vidav [9], it follows that (1.2) if T is hermitian under the equivalent norm || || on X, then ||β ίrΓ || = 1 for all real r. (1.3) Supposed i = 1, 2, ••, w are finitely many commuting families of operators on a Banach space X, each of them being hermitian-equivalent and F = \Jΐ =1 F { is also a commuting family. Then the family F is hermitian-equivalent.…”

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confidence: 99%

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Unitary operators in Banach spaces

Panchapagesan¹

1967

Pacific J. Math.

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“…Combining the results of Vidav [8], Berkson [1], and Glickfeld [6] we obtain the result that if A is a unital Banach algebra such that A H(A)+iH(A) then A is a B*-algebra under the Vidav-involution. Here, we extend this result to the nonunital case in the form of Lemma 3.1.…”

mentioning

confidence: 86%