On the separation properties of
Malgorzata Wójcicka (1986)
Colloquium Mathematicae
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Malgorzata Wójcicka (1986)
Colloquium Mathematicae
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Yoichi Uetake (2001)
Studia Mathematica
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Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product . For b, c ∈ X, a weak resolvent of A is the complex function of the form . We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.
Hisao Kato (2015)
Commentationes Mathematicae Universitatis Carolinae
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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces and , where and denote the Hilbert cube and a Cantor set, respectively.
Ryszard Grząślewicz (1981)
Colloquium Mathematicae
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H. G. Dales, M. Daws, H. L. Pham, P. Ramsden
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The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show...
Ryszard Grząślewicz (1988)
Colloquium Mathematicae
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Sunanda Naik, Karabi Rajbangshi (2015)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator by , where a and b are non-negative real numbers. In particular, for a = b = β, becomes the generalized Hilbert operator , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that is bounded on Dirichlet-type spaces , 0 < p < 2, and on Bergman spaces , 2 < p < ∞. Also we find an upper bound for the norm of the operator ....
Stanisław Kwapień, Jan Mycielski (2001)
Studia Mathematica
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The Kaczmarz algorithm of successive projections suggests the following concept. A sequence of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and , where . We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.
Daniel Li (1995)
Colloquium Mathematicae
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Monojit Bhattacharjee, Jaydeb Sarkar (2016)
Studia Mathematica
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We study analytic models of operators of class with natural positivity assumptions. In particular, we prove that for an m-hypercontraction on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that and , where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their...
Piotr Niemiec (2012)
Bulletin of the Polish Academy of Sciences. Mathematics
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Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α ≥ ℵ₀, A is a closed set in X and f: X → M is a map into a manifold M modelled on a Hilbert space of dimension α such that , then for every open cover of M there...
Antonio Avilés, Witold Marciszewski (2015)
Studia Mathematica
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We study extension operators between spaces of continuous functions on the spaces of subsets of X of cardinality at most n. As an application, we show that if is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator .
Maria Nowak, Renata Rososzczuk (2014)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces , . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.
Riad Masri (2013)
Annales de l’institut Fourier
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We generalize Kronecker’s solution of Pell’s equation to CM fields whose Galois group over is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of . Assuming Schanuel’s conjecture, we show that when has degree greater than 2 over these CM values...
Tadeusz Dobrowolski, Witold Marciszewski (2002)
Fundamenta Mathematicae
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In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an -subset of X and contains a retract R so that is not homeomorphic to . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.
Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)
Annales de l’institut Fourier
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A variety over a field is of Hilbert type if is not thin. We prove that if is a dominant morphism of -varieties and both and all fibers , , are of Hilbert type, then so is . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.
Daniel Azagra, Mar Jiménez-Sevilla (2002)
Bulletin de la Société Mathématique de France
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We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth...
G. Pantsulaia (2004)
Bulletin of the Polish Academy of Sciences. Mathematics
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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift of μ by a vector are neither equivalent nor orthogonal. This extends a result established in [7].
B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)
Studia Mathematica
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We determine the norm in , 1 < p < ∞, of the operator , where and are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real...
Gilles Pisier (2014)
Journal of the European Mathematical Society
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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of -spaces needed to represent (up to a constant ) the -version of the -dimensional operator Hilbert space as a direct sum of copies of . We show that, when is close to 1, this multiplicity grows...
Chia-chi Tung (2013)
Annales Polonici Mathematici
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Conditions characterizing the membership of the ideal of a subvariety arising from (effective) divisors in a product complex space Y × X are given. For the algebra of relative regular functions on an algebraic variety V, the strict stability is proved, in the case where Y is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in , respectively,...
Teresa Bermúdez, Carlos Díaz Mendoza, Antonio Martinón (2012)
Studia Mathematica
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A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, . We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if and are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if is an (m,p)-isometry and is an (l,p)-isometry, then is an (h,p)-isometry, where t = gcd(r,s)...
Adam Osękowski (2012)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let df be a Hilbert-space-valued martingale difference sequence. The paper is devoted to a new, elementary proof of the estimate with as p → ∞.
Piotr Niemiec (2012)
Studia Mathematica
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For a linear operator T in a Banach space let denote the point spectrum of T, let for finite n > 0 be the set of all such that dim ker(T - λ) = n and let be the set of all for which ker(T - λ) is infinite-dimensional. It is shown that is , is and for each finite n the set is the intersection of an set and a set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...
Félix Cabello Sánchez (1999)
Studia Mathematica
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Let X be a normed space and the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if acts transitively on the unit sphere then X must be an inner product space.