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Complete Pick positivity and unitary invariance

Angshuman Bhattacharya, Tirthankar Bhattacharyya (2010)

Studia Mathematica

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel k S ( z , w ) = ( 1 - z w ̅ ) - 1 for |z|,|w| < 1, by means of ( 1 / k S ) ( T , T * ) ≥ 0 , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful...

Completely monotone functions of finite order and Agler's conditions

Sameer Chavan, V. M. Sholapurkar (2015)

Studia Mathematica

Motivated by some structural properties of Drury-Arveson d-shift, we investigate a class of functions consisting of polynomials and completely monotone functions defined on the semi-group ℕ of non-negative integers, and its operator-theoretic counterpart which we refer to as the class of completely hypercontractive tuples of finite order. We obtain a Lévy-Khinchin type integral representation for the spherical generating tuples associated with such operator tuples and discuss its applications.

Complétude des noyaux reproduisants dans les espaces modèles

Emmanuel Fricain (2002)

Annales de l’institut Fourier

Soit ( λ n ) n ≥ 1 une suite de Blaschke du disque unité 𝔻 et Θ une fonction intérieure. On suppose que la suite de noyaux reproduisants k Θ ( z , λ n ) : = 1 - Θ ( λ n ) ¯ Θ ( z ) 1 - λ n ¯ z n ≥ 1 est complète dans l’espace modèle K Θ p : = H p ∩ Θ H 0 p ¯ , 1 &lt; p &lt; + ∞ . On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences ( λ n ) n ≥ 1 mais également sous l’effet de perturbations de la fonction Θ . On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite ...

Composition operators and the Hilbert matrix

E. Diamantopoulos, Aristomenis Siskakis (2000)

Studia Mathematica

The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.

Computing the numerical range of Krein space operators

Natalia Bebiano, J. da Providência, A. Nata, J.P. da Providência (2015)

Open Mathematics

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined...

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