Displaying similar documents to “Addendum to “On Hilbert sets and C λ ( g ) -spaces with no subspace isomorphic to c 0 ””

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

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 ⟨ · , · ⟩ X . For b, c ∈ X, a weak resolvent of A is the complex function of the form ⟨ ( I - z A ) - 1 b , c ⟩ X . 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.

On isometrical extension properties of function spaces

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 C ( Q ) and C ( Δ ) , where Q and Δ denote the Hilbert cube [ 0 , 1 ] ∞ and a Cantor set, respectively.

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

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A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and A 1 / 2 . It also depends on a certain angle between A() and the orthogonal of .

Generalized atomic subspaces for operators in Hilbert spaces

Prasenjit Ghosh, Tapas Kumar Samanta (2022)

Mathematica Bohemica

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We introduce the notion of a g -atomic subspace for a bounded linear operator and construct several useful resolutions of the identity operator on a Hilbert space using the theory of g -fusion frames. Also, we shall describe the concept of frame operator for a pair of g -fusion Bessel sequences and some of their properties.

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

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The structure of the closed linear span of the Rademacher functions in the Cesàro space C e s ∞ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in C e s ∞ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of C e s ∞ if 1 < p < ∞.

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

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 a , b by a , b ( f ) ( z ) = Γ ( a + 1 ) / Γ ( b + 1 ) ∫ 0 1 ( f ( t ) ( 1 - t ) b ) / ( ( 1 - t z ) a + 1 ) d t , where a and b are non-negative real numbers. In particular, for a = b = β, a , b becomes the generalized Hilbert operator β , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that a , b is bounded on Dirichlet-type spaces S p , 0 < p < 2, and on Bergman spaces A p , 2 < p < ∞. Also we find an upper bound for the norm of the operator a , b ....

Hilbert series of the Grassmannian and k -Narayana numbers

Lukas Braun (2019)

Communications in Mathematics

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We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the q -Hilbert series is a Vandermonde-like determinant. We show that the h -polynomial of the Grassmannian coincides with the k -Narayana polynomial. A simplified formula for the h -polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the k -Narayana numbers,...

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let σ p ( T ) denote the point spectrum of T, let σ p , n ( T ) for finite n > 0 be the set of all λ ∈ σ p ( T ) such that dim ker(T - λ) = n and let σ p , ∞ ( T ) be the set of all λ ∈ σ p ( T ) for which ker(T - λ) is infinite-dimensional. It is shown that σ p ( T ) is ℱ σ , σ p , ∞ ( T ) is ℱ σ δ and for each finite n the set σ p , n ( T ) is the intersection of an ℱ σ set and a δ 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...

On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

Stanisław Kwapień, Jan Mycielski (2001)

Studia Mathematica

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The Kaczmarz algorithm of successive projections suggests the following concept. A sequence ( e k ) 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 x ₙ = x n - 1 + α ₙ e ₙ , where α ₙ = ⟨ x - x n - 1 , e ₙ ⟩ . 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.

Best constants for some operators associated with the Fourier and Hilbert transforms

B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)

Studia Mathematica

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We determine the norm in L p ( ℝ ₊ ) , 1 < p < ∞, of the operator I - ℱ s ℱ c , where ℱ c and ℱ s 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 L p -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real...

Some equivalent metrics for bounded normal operators

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2018)

Mathematica Bohemica

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Some stronger and equivalent metrics are defined on ℳ , the set of all bounded normal operators on a Hilbert space H and then some topological properties of ℳ are investigated.

Failure of the Factor Theorem for Borel pre-Hilbert spaces

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 F σ δ σ -subset of X and contains a retract R so that R × E ω is not homeomorphic to E ω . This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

Rosenthal operator spaces

M. Junge, N. J. Nielsen, T. Oikhberg (2008)

Studia Mathematica

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In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L p -space, then it is either an L p -space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative L p -spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which...

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety X over a field K is of Hilbert type if X ( K ) is not thin. We prove that if f : X → S is a dominant morphism of K -varieties and both S and all fibers f - 1 ( s ) , s ∈ S ( K ) , are of Hilbert type, then so is X . 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.

Kronecker’s solution of Pell’s equation for CM fields

Riad Masri (2013)

Annales de l’institut Fourier

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We generalize Kronecker’s solution of Pell’s equation to CM fields K whose Galois group over ℚ 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 K is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of K . Assuming Schanuel’s conjecture, we show that when K has degree greater than 2 over ℚ these CM values...

A Note on the Burkholder-Rosenthal Inequality

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 ∥ ∑ k = 0 ∞ d f k ∥ p ≤ C p ∥ ( ∑ k = 0 ∞ ( | d f k | ² | ℱ k - 1 ) ) 1 / 2 ∥ p + ∥ ( ∑ k = 0 ∞ | d f k | p ) 1 / p ∥ p , with C p = O ( p / l n p ) as p → ∞.

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form C Λ ( G ) or L ¹ Λ ( G ) and which quotients of the form C ( G ) / C Λ ( G ) or L ¹ ( G ) / L ¹ Λ ( G ) have the Daugavet property. We show that C Λ ( G ) is a rich subspace of C(G) if and only if Γ ∖ Λ - 1 is a semi-Riesz set. If L ¹ Λ ( G ) is a rich subspace of L¹(G), then C Λ ( G ) is a rich subspace of C(G) as well. Concerning quotients, we prove that C ( G ) / C Λ ( G ) has the Daugavet property if Λ is a Rosenthal set, and that L ¹ Λ ( G ) is a poor subspace of L¹(G)...

On the Hilbert 2 -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini, Idriss Jerrari (2021)

Czechoslovak Mathematical Journal

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Let 𝕜 = ℚ 2 , d be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and 𝕜 2 ( 2 ) its second Hilbert 2 -class field. Denote by G = Gal ( 𝕜 2 ( 2 ) / 𝕜 ) the Galois group of 𝕜 2 ( 2 ) / 𝕜 . The purpose of this note is to investigate the Hilbert 2 -class field tower of 𝕜 and then deduce the structure of G .