On a function that realizes the maximal spectral type

Krzysztof Frączek

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Displaying similar documents to “On a function that realizes the maximal spectral type”

Spectral decompositions and harmonic analysis on UMD spaces

Earl Berkson, T. Gillespie (1994)

Studia Mathematica

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We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_{X}^{p}$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X. ...

On some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^{p} (ℝ)$ -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m} f (x) = p . v . \int k (x - y) m (x + y) f (y) d y$ where $k (x) = \sum_{j \in ℤ} 2^{j} φ_{j} (2^{j} x)$ for a family of functions ${φ_{j}}_{j \in ℤ}$ satisfying conditions (1.1)-(1.3) given below.

On spectral representation for selfadjoint operators. Expansion in generalized eigenelements

Eberhard Gerlach (1965)

Annales de l'institut Fourier

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L’auteur reprend l’étude classique de la représentation spectrale d’un opérateur auto-adjoint $A$ dans un espace de Hilbert $ℋ$ . Il y ajoute des précisions nouvelles qui conduisent à la définition du projecteur infinitésimal $P^{) λ)}$ sur l’espace des vecteurs propres généralisés $ℋ^{(λ)}$ . Il obtient, par conséquent, des énoncés plus précis de bien des théorèmes classiques. Il introduit ensuite la notion de “ $A$ -expansibilité” d’un sous-ensemble $S \subset ℋ$ . Cette notion est appliquée à l’étude des espaces fonctionnels...

On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{ξ}$ generated by subsequences $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ , respectively, of the natural Schauder basis $(e_{n}^{ξ})$ of $X^{ξ}$ are isomorphic if and only if $(e_{l_{n}}^{ξ})$ and $(e_{m_{n}}^{ξ})$ are equivalent. Further, $X^{ξ}$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $(e_{n}^{ξ})$ . It is also shown that there exists a complemented subspace spanned by a block basis of $(e_{n}^{ξ})$ , which is not isomorphic to a subspace generated by a subsequence of $(e_{n}^{ζ})$ ,...

Spectral dilation of operator-valued measures and its application to infinite-dimensional harmonizable processes

A. Makagon, H. Salehi (1987)

Studia Mathematica

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A lifting theorem for locally convex subspaces of $L_{0}$

R. Faber (1995)

Studia Mathematica

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We prove that for every closed locally convex subspace E of $L_{0}$ and for any continuous linear operator T from $L_{0}$ to $L_{0} / E$ there is a continuous linear operator S from $L_{0}$ to $L_{0}$ such that T = QS where Q is the quotient map from $L_{0}$ to $L_{0} / E$ .

On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát (1998)

Studia Mathematica

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Let G be the Walsh group. For $f \in L^{1} (G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ * f ≔ s u p_{n} | σ_{n} f | .$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥ σ * f ∥_{1} \leq c ∥ | f | ∥_{H}$ , where H is the Hardy space on the Walsh group.

On the representation of functions by orthogonal series in weighted $L^{p}$ spaces

M. Grigorian (1999)

Studia Mathematica

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It is proved that if $φ_{n}$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form $\sum_{k = 1}^{\infty} c_{k} φ_{k} (x)$ , where $c_{k} \in l_{q}$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1] = f : ʃ_{0}^{1} {| f (x) |}^{p} μ (x) d x < \infty$ there are numbers $ɛ_{k}$ , k=1,2,…, $ɛ_{k}$ = 1 or 0, such that $l i m_{n \to \infty} ʃ_{0}^{1} {| \sum_{k = 1}^{n} ɛ_{k} c_{k} φ_{k} (x) - f (x) |}^{p} μ (x) d x = 0 .$ 2. For every p ∈ [1,2) and $f \in L_{μ}^{p} [0, 1]$ there are a function $g \in L^{1} [0, 1]$ with g(x) = f(x) on E and numbers $δ_{k}$ , k=1,2,…, $δ_{k} = 1$ or 0,...

On invariant measures for power bounded positive operators

Ryotaro Sato (1996)

Studia Mathematica

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We give a counterexample showing that $\bar{(I - T *) L_{\infty}} \cap L_{\infty}^{+} = 0$ does not imply the existence of a strictly positive function u in $L_{1}$ with Tu = u, where T is a power bounded positive linear operator on $L_{1}$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.

Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0, 1}$ into itself

G. Sampson (1993)

Studia Mathematica

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We consider operators of the form $(Ω f) (y) = ʃ_{- \infty}^{\infty} Ω (y, u) f (u) d u$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h \in L^{\infty}$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ_{1}^{0, 1}$ (= B) into itself. In particular, all operators with $h (y) = e^{{i | y |}^{a}}$ , a > 0, a ≠ 1, map B into itself.

Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz (1994)

Studia Mathematica

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$(s_{j k} : j, k = 0, 1, . . .)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $(s_{j k})$ converges in Pringsheim’s sense. These conditions are satisfied if $(s_{j k})$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $(s_{j k})$ is summable (C,1,1) to a finite limit and there exist constants $n_{1} > 0$ and H such that $j k (s_{j k} - s_{j - 1, k} - s_{j - 1, k} + s_{j - 1, k - 1}) \geq - H$ , $j (s_{j k} - s_{j - 1, k}) \geq - H$ and $k (s_{j k} - s_{j, k - 1}) \geq - H$ whenever $j, k > n_{1}$ , then...

Semi-spectral integrals and related mappings

J. Janas (1973)

Studia Mathematica

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A continuous real-valued function on $E^{n}$ almost everywhere 1-1

J. Denny (1964)

Fundamenta Mathematicae

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