Uniform convergence of operators onL ? and similar spaces

Lotz, Heinrich P.

doi:10.1007/bf01160459

Cited by 90 publications

(85 citation statements)

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“…The space H ∞ 1 (C) is a Grothendieck Banach space with the Dunford-Pettis property since it is isomorphic to ∞ by Galbis [11] or Lusky [20]. As ||J n || 1 /n → 0, we can apply [17,Theorem 8] or [18,Theorem 5] to conclude that J is not mean ergodic in H ∞ 1 (C) because it is not uniformly mean ergodic by Theorem 3.5 (iv) and Proposition 2.3.…”

Section: Proposition 33 Let V Be a Weight Such That J Is Continuousmentioning

confidence: 99%

Classical operators on weighted Banach spaces of entire functions

Beltrán¹,

Bonet²,

Fernández

2013

Proc. Amer. Math. Soc.

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Section: Proposition 33 Let V Be a Weight Such That J Is Continuousmentioning

confidence: 99%

Classical operators on weighted Banach spaces of entire functions

Beltrán¹,

Bonet²,

Fernández

2013

Proc. Amer. Math. Soc.

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“…(Note that Lemma 5 is a special case of the result of [3] and [12]; if the ideal is zero, that is to say, if pAp is an abelian AW*-algebra, the proof of Proposition Al of [11] is valid.) Hence, \\<p a -l\\-*• 0.…”

Section: Upmentioning

confidence: 99%

“…We give a proof which is valid for quotients (unlike that of [11]) which is different from that of [3] and [12]. It is sufficient to prove that sup{||^ne -e\\; ea projection in A} ->• 0.…”

Section: It Follows Thatmentioning

confidence: 99%

“…While the proof of Kishimoto and Robinson does not extend to the case of a quotient of a W*-algebra, a different proof, given by Coulhon in [3] and by Lotz in [12], does extend in this way. As a result-as the rest of the methods pass to quotients-the full result holds for any quotient of a W*-algebra (or AW*-algebra).…”

mentioning

confidence: 98%

“…(Actually, all that they use is the Choi-Schwarz inequality, which holds for 2-positive maps; see [12,Corollary 2.8].) The purpose of the present note is to extend these results to the missing case, namely, the case that the algebra is finite but not abelian.…”

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confidence: 99%

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On the convergence of a sequence of completely positive maps to the identity

Elliott¹

2000

J Aust Math Soc A

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Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

Okada

1988

Mathematische Nachrichten

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I t is well known that every non-empty closed subset of the complex plane is the spectrum of a normal operator in a HUBERT space. This result cannot be extended to the case of BANACH spaces. Indeed, it is shown in [17] that the spectrum of a spectral operator in a GR~THENDIECX space with the DUNFORD-PETTIS property is a finite set; the argument is based on the fact that such a BANACH space has no SCHAUDER decomposition. The purpose of this note is to show that, in a locally convex spice, the existence of an unconditional SCHAUDER decomposition ensures the property that every non-empty closed subset of the complex plane is the spectrum of a scalar-type spectral operator (see Theorem 6). As a consequence, a BANACH space has the above property if and only if it has an unconditional SCHAUDER decomposition (see Corollary 7).Throughout this note, let X be a sequentially complete, locally convex &us-DORFP space and X' its dual space. Let L ( X ) denote the space of all continuous linear operators from X into itself. If the space L ( X ) is equipped with the topology of pointwise convergence on X, then it will be denoted by L , ( X ) . The identity operator on the space X will be denoted by I . Let T be a linear operator defined on a subspace of X with values in X. Suppose that 1 is a complex number such that the operator 1I -T is an injection with dense range whose inverse (U -T)-1 defined on the range of 1I -T is extendable to a continuous operator from X into itself. Then, the extension of (11-T)-1 to X will be denoted by R (1.; T). The resolvent equation R(1; T)-R(C; T ) = -( I -[ ) R ( I ; T ) R([; T )holds whenever I and i are complex numbers for which the operators R(A; T ) and R(C; T ) exist.The resolvent set e(T) of a linear operator T from a subspace of X into X consists of those complex numbers 1 such that there is a neighbourhood U of 1 in the

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Uniform convergence of operators onL ? and similar spaces

Cited by 90 publications

References 7 publications

Classical operators on weighted Banach spaces of entire functions

Classical operators on weighted Banach spaces of entire functions

On the convergence of a sequence of completely positive maps to the identity

Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

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