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Operators preserving orthogonality of polynomials

Francisco Marcellán, Franciszek Szafraniec (1996)

Studia Mathematica

Let S be a degree preserving linear operator of ℝ[X] into itself. The question is if, preserving orthogonality of some orthogonal polynomial sequences, S must necessarily be an operator of composition with some affine function of ℝ. In [2] this problem was considered for S mapping sequences of Laguerre polynomials onto sequences of orthogonal polynomials. Here we improve substantially the theorems of [2] as well as disprove the conjecture proposed there. We also consider the same questions for polynomials...

Operators whose adjoints are quasi p-nuclear

J. M. Delgado, C. Piñeiro, E. Serrano (2010)

Studia Mathematica

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with K ⊆ ∑ ₙ α ₙ x ₙ : ( α ₙ ) ∈ B ℓ p ' . We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

Operators with absolute continuity properties: an application to quasinormality

Zenon Jan Jabłoński, Il Bong Jung, Jan Stochel (2013)

Studia Mathematica

An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerge in this context are found. Various examples and counterexamples illustrating the concepts of the paper are constructed by using weighted shifts on directed trees. Generalizations of these results that cover the case of q-quasinormal operators are established.

Operators with hypercyclic Cesaro means

Fernando León-Saavedra (2002)

Studia Mathematica

An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit T ⁿ x n ≥ 1 is dense in ℬ. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ℬ. If the arithmetic means of the orbit of x are dense in ℬ then the operator T is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector...

Optimal domains for kernel operators on [0,∞) × [0,∞)

Olvido Delgado (2006)

Studia Mathematica

Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are based on the relation between linear operators and vector measures.

Optimal domains for the kernel operator associated with Sobolev's inequality

Guillermo P. Curbera, Werner J. Ricker (2003)

Studia Mathematica

Refinements of the classical Sobolev inequality lead to optimal domain problems in a natural way. This is made precise in recent work of Edmunds, Kerman and Pick; the fundamental technique is to prove that the (generalized) Sobolev inequality is equivalent to the boundedness of an associated kernel operator on [0,1]. We make a detailed study of both the optimal domain, providing various characterizations of it, and of properties of the kernel operator when it is extended to act in its optimal domain....

Orbits in symmetric spaces, II

N. J. Kalton, F. A. Sukochev, D. Zanin (2010)

Studia Mathematica

Suppose E is fully symmetric Banach function space on (0,1) or (0,∞) or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on f ∈ E so that its orbit Ω(f) is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.

Orbits under a class of isometries of L¹[0,1]

Terje Hõim (2004)

Studia Mathematica

We study the orbits of isometries of L¹[0,1]. For a certain class of isometries we show that the set of functions f in L¹[0,1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis e₁,e₂,e₃,... of l¹ is an open dense set. In the proof we develop a new method to get copies of l¹ inside L¹[0,1] using geometric progressions. This method does not use disjoint or relatively disjoint supports, which seems to be the most common way to get such copies. We also use this...

Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Nizar Jaoua (1999)

Studia Mathematica

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces H p 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with X = H p or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into L h . We give a complete characterization...

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