Numerical ranges of nilpotent operators

For any operator A on a Hilbert space, let W(A), w(A) and w₀(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ=0, then w(A)?(n-1)w₀(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w₀(A), (2) A is unitarily equivalent to an operator of the form aA_n⊕A^′, where a is a scalar satisfying |a|=2w₀(A), A_n is the n-by-n matrix

     

Numerical ranges of Toeplitz operators with matrix symbols

For Toeplitz operators acting on the vector Hardy space $H^2$ with definite or indefinite metric, the closure of the respective numerical range is completely described. In the definite case, some observations regarding its boundary are also made.     

Numerical ranges of weighted composition operators

The operator that takes the function f   to ψf°φ$\psi f°\phi$ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H²$H^2$, the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.     

A duality for Boolean algebras with operators

Jónsson and Tarski's extension and representation theorems for Boolean algebras with operators ([7], p. 926 and p. 933) can be extended to homomorphisms between these algebras. The result obtained takes the form of a duality between the category of Boolean algebras with operators and that of the “algebras in the wider sense” (whose subjects are defined in [7]) with a suitable topology. This duality generalizes results of Pierce ([10], p. 38). Moreover, it can be extended to more general objects such as Boolean algebras with non-normal operators and even to arbitrary distributive lattices with operators.     

Functional calculus and duality for closed operators

In this paper, we continue our spectral-theoretic study [8] of unbounded closed operators in the framework of the spectral decomposition property and decomposable operators. Given a closed operator T with nonempty resolvent set, let f → f(T) be the homomorphism of the functional calculus. We show that if T has the spectral decomposition property, then f(T) is decomposable. Conversely, if f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable implies that T has the spectral decomposition property. A spectral duality theorems follows as a corollary. Furthermore, we obtain an analytic-type property for the canonical embedding J of the underlying Banach space X into its second dual $\text{X}{}^7$.     

Classes of Hardy spaces associated with operators, duality theorem and applications

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space and a space associated with the operator were introduced and studied. In this paper we define a class of spaces associated with the operator for a range of acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical spaces. We then establish a duality theorem between the spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on and give the inclusion between the classical spaces and the spaces associated with operators.

     

Koplienko–Neidhardt trace formula for pairs of contraction operators and pairs of maximal dissipative operators

We present a generalization of Koplienko–Neidhardt trace formula for pairs of Hilbert space operators (T , V ) with T contractive and V unitary such that T – V is a Hilbert–Schmidt operator. We extend the result to pairs of contractions and then, via Cayley transform, to pairs of maximal dissipative operators. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximating curves for non-expansive mappings and accretive operators

In this article, under the hypotheses that E is a reflexive Banach space which has a weakly continuous duality mapping J _? with gauge function ?, we investigate several alternative viscosity approximation methods for finding some common fixed points of infinite non-expansive mappings, and prove two strong convergence theorems.     

Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings

Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range.

     

On the duality for linear operators and distribution semi-groups

Sommario La dualità per uno spazio di Banach X è studiata in relazione a una arbitraria famiglia di operatori lineari continui di X. La teoria generale è applicata allo studio del duale di un semi-gruppo distribuzione.

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Research supported at an earlier stage by CNR (Gruppo 46) and later by OSR while the author was a visitor at Massachusetts Institute of Technology.     

Local duality of Hecke operators for symplectic and orthogonal groups

A formula for the duality of Hecke operators for symplectic and orthogonal groups on spaces of local Weil representations is obtained in terms of spherical functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 37–59, 1990.     

Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. Received: 10 August 2007