A classification of projectors

Gustavo Corach; Alejandra Maestripieri; Demetrio Stojanoff

Displaying similar documents to “A classification of projectors”

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.

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.

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 $⟨ \cdot, \cdot ⟩_{X}$ . 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.

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 $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , 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 $_{a, b}$ ....

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

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We study extension operators between spaces of continuous functions on the spaces $σ ₙ (2^{X})$ of subsets of X of cardinality at most n. As an application, we show that if $B_{H}$ 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 $T : C (λ B_{H}) \to C (μ B_{H})$ .

On the separation properties of $K_{ω}$

Malgorzata Wójcicka (1986)

Colloquium Mathematicae

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The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

Piotr Niemiec

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An ideal of N-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $C D D_{N}$ of all unitary equivalence classes of such N-tuples...

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators $T$ for which there exists a fundamental decomposition $𝕂 = 𝕂^{+} \oplus 𝕂^{-}$ of the Krein space $𝕂$ with $𝕂^{+}$ and $𝕂^{-}$ invariant under $T$ .

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

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 ${[0, 1]}^{\infty}$ and a Cantor set, respectively.

Completely 1-complemented subspaces of the Schatten spaces $S^{p}$

Jean Roydor (2007)

Banach Center Publications

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We describe the subspaces of $S^{p}$ (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $Δ_{A, B}$ with respect to the wider class of unitarily invariant...

Multiplication operators on $L (L_{p})$ and $ℓ_{p}$ -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on $L (L_{p}),$ 1<p< $, i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s, a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l$ p $- s t r i c t l y s i n g u l a r o p e r a t o r s o n$ Lp $. T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r$ T $o n$ Lp $,$ 1<p<2 $, i s$ p $- s t r i c t l y s i n g u l a r a n d$ T|X $i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e$ X $o f$ Lp $, t h e n$ X $e m b e d s i n t o$ Lr $f o r a l l$ r<2 $, b u t$ X $n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e .$ It is also shown that if $T$ is convolution by a biased coin on $L_{p}$ of the Cantor group, $1 \leq p < 2$ , and $T_{| X}$ is an isomorphism for some reflexive subspace $X$ of $L_{p}$ , then $X$ is isomorphic to a Hilbert space. The case $p = 1$ answers a question asked by Rosenthal in 1976.

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

Best approximation in spaces of bounded linear operators

Grzegorz Lewicki

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CONTENTSChapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9Chapter...

Linear maps preserving elements annihilated by the polynomial $X Y - Y X^{†}$

Jianlian Cui, Jinchuan Hou (2006)

Studia Mathematica

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Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $Φ (A) Φ (B) = Φ (B) Φ {(A)}^{†}$ for all A, B ∈ ℬ(H) with $A B = B A^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $Φ (A) = c U A U^{†}$ for all A ∈ ℬ(H).

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

Hermitian operators on $H_{E}^{\infty}$ and $S_{}^{\infty}$

James Jamison (2013)

Colloquium Mathematicae

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A complete characterization of bounded and unbounded norm hermitian operators on $H_{E}^{\infty}$ is given for the case when E is a complex Banach space with trivial multiplier algebra. As a consequence, the bi-circular projections on $H_{E}^{\infty}$ are determined. We also characterize a subclass of hermitian operators on $S_{}^{\infty}$ for a complex Hilbert space.

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

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Given two n-tuples $a = (a_{1}, . . ., a_{n})$ and $b = (b_{1}, . . ., b_{n})$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E a_{j} = b_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^{n}$ .