scholarly journals Operator self-similar processes on Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Mihaela T. Matache ◽  
Valentin Matache
Keyword(s):  
Hilbert Space ◽  
Stochastic Processes ◽  
Banach Spaces ◽  
Linear Space ◽  
Power Function ◽  
Linear Subspace ◽  
Multiplicative Group ◽  
The Family ◽  
Self Similar ◽  
Dense Linear Subspace

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form {sH:s>0}, if the expectations of the process span a dense linear subspace of category 2. The existence of a scaling family of the form {sH:s>0} is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Some properties of D-weak operator topology

2020 ◽  
Vol 70 (3) ◽  
pp. 753-758
Author(s):  
Marcel Polakovič
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Linear Subspace ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Weak Operator Topology ◽  
Generalized Effect Algebra ◽  
Weak Operator ◽  
Dense Linear Subspace

AbstractLet 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

scholarly journals On the problem of non-smoothness of non-reflexive second conjugate spaces

1975 ◽  
Vol 12 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Canonical Embedding ◽  
Reflexive Banach Spaces ◽  
Closed Linear Subspace ◽  
Conjugate Space ◽  
The Family

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

Parareductive Operators on Banach Spaces

1994 ◽  
Vol 37 (3) ◽  
pp. 346-350
Author(s):  
Roman Drnovšek
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Space Extension ◽  
Space Result

AbstractThis note gives a Banach space extension of the Hilbert space result due to P. A. Fillmore (see [3]). In particular, it is shown that the adjoint T* = A — iB of an operator T = A + iB (with A and B hermitian) is a polynomial in T if and only if T* leaves invariant every linear subspace invariant under T, and this is equivalent to the assertion that T* leaves invariant every paraclosed subspace invariant under T.

GENERALIZATION OF THE FRAME OPERATOR AND THE CANONICAL DUAL FRAME TO BANACH SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 631-643 ◽  
Author(s):  
Diana T. Stoeva
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Dual Frame ◽  
Frame Operator ◽  
Hilbert Frames ◽  
The Family

Xd-frames for Banach spaces are generalization of Hilbert frames. In this paper we extend the concepts of frame operator and canonical dual to the case of Xd-frames. For a given Xd-frame {gi} for the Banach space X we define an Xd-frame map𝕊 : X → X* and determine conditions, which imply that 𝕊 is invertible and the family {𝕊-1gi} is an [Formula: see text]-frame for X* such that f = ∑gi(f)𝕊-1gi for every f ∈ X and g = ∑g(𝕊-1gi)gi for every g ∈ X*. If X is a Hilbert space and {gi} is a frame for X, then the ℓ2-frame map 𝕊 gives the frame operator S and the family {𝕊-1gi} coincides with the canonical dual of {gi}.

scholarly journals Fixed points of quasi-nonexpansive mappings

1972 ◽  
Vol 13 (2) ◽  
pp. 167-170 ◽  
Author(s):  
W. G. Dotson
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Linear Space ◽  
Complex Plane ◽  
Normed Linear Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
The Other ◽  
Commuting Mappings ◽  
Analytic Mappings

A self-mapping T of a subset C of a normed linear space is said to be non-expansive provided ║Tx — Ty║ ≦ ║x – y║ holds for all x, y ∈ C. There has been a number of recent results on common fixed points of commutative families of nonexpansive mappings in Banach spaces, for example see DeMarr [6], Browder [3], and Belluce and Kirk [1], [2]. There have also been several recent results concerning common fixed points of two commuting mappings, one of which satisfies some condition like nonexpansiveness while the other is only continuous, for example see DeMarr [5], Jungck [8], Singh [11], [12], and Cano [4]. These results, with the exception of Cano's, have been confined to mappings from the reals to the reals. Some recent results on common fixed points of commuting analytic mappings in the complex plane have also been obtained, for example see Singh [13] and Shields [10].

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Loudness Growth under Masking: Relation to True Sensorineural Impairment

1973 ◽  
Vol 16 (4) ◽  
pp. 597-607 ◽  
Author(s):  
Alan M. Richards
Keyword(s):  
Hearing Loss ◽  
Power Function ◽  
Threshold Shift ◽  
Essential Difference ◽  
Frequency Dependent ◽  
Masker Level ◽  
Loudness Recruitment ◽  
The Family ◽  
Balance Functions ◽  
Loudness Growth

Alternate binaural loudness balances between masked and unmasked normal ears were performed to examine the growth of loudness as a function of masker level at each of several frequencies (500, 1000, 2000, and 4000 Hz) and to determine whether the recruitmentlike phenomenon in masked ears is comparable in its growth and form to actual recruitment growth in sensorineural impaired ears. The results for 28 subjects indicated that for all frequencies a power function relating the perceived loudness in the masked ear to the unmasked ear could be drawn, and that the slope of this function rose as a function of increased masking. The family of slopes for each frequency was linearly related to the induced threshold shift. The slope of this latter relation proved to be frequency dependent. Comparison between the slope growth in simulated hearing loss and the family of loudness-balance slopes obtained from patients with true unilateral loss of varying degree indicated that the slopes of loudness-balance functions in the latter group also increased linearly with increased loss. In this latter instance, however, the slope growth was not frequency dependent, thus pointing to an essential difference between simulated and actual loudness recruitment growth.

Extremal Properties of an Intermittent Poisson Process Generating 1/f Noise

2016 ◽  
Vol 15 (04) ◽  
pp. 1650028 ◽  
Author(s):  
Ferdinand Grüneis
Keyword(s):  
Stochastic Processes ◽  
Poisson Process ◽  
Total Power ◽  
Spectral Features ◽  
Active Period ◽  
Quiescent Period ◽  
The Family ◽  
Random Intervals ◽  
Theoretical Results ◽  
Extremal Properties

It is well-known that the total power of a signal exhibiting a pure [Formula: see text] shape is divergent. This phenomenon is also called the infrared catastrophe. Mandelbrot claims that the infrared catastrophe can be overcome by stochastic processes which alternate between active and quiescent states. We investigate an intermittent Poisson process (IPP) which belongs to the family of stochastic processes suggested by Mandelbrot. During the intermission [Formula: see text] (quiescent period) the signal is zero. The active period is divided into random intervals of mean length [Formula: see text] consisting of a fluctuating number of events; this is giving rise to so-called clusters. The advantage of our treatment is that the spectral features of the IPP can be derived analytically. Our considerations are focused on the case that intermission is only a small disturbance of the Poisson process, i.e., to the case that [Formula: see text]. This makes it difficult or even impossible to discriminate a spike train of such an IPP from that of a Poisson process. We investigate the conditions under which a [Formula: see text] spectrum can be observed. It is shown that [Formula: see text] noise generated by the IPP is accompanied with extreme variance. In agreement with the considerations of Mandelbrot, the IPP avoids the infrared catastrophe. Spectral analysis of the simulated IPP confirms our theoretical results. The IPP is a model for an almost random walk generating both white and [Formula: see text] noise and can be applied for an interpretation of [Formula: see text] noise in metallic resistors.

scholarly journals Characterizing Complete Erdős Space

2009 ◽  
Vol 61 (1) ◽  
pp. 124-140 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jan van Mill
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Convergent Sequence ◽  
Cantor Set

Abstract. The space now known as complete Erdős space was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space ℓ2 consisting of all vectors such that every coordinate is in the convergent sequence ﹛0﹜ ∪ ﹛1/n : n ∈ℕ﹜. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of . As an application we determine the class of factors of . In another application we determine precisely which of the spaces that can be constructed in the Banach spaces ℓp according to the ‘Erdős method’ are homeomorphic to . A novel application states that if I is a Polishable Fσ-ideal on ω, then I with the Polish topology is homeomorphic to either ℤ, the Cantor set 2ω, ℤ × 2ω, or . This last result answers a question that was asked by Stevo Todorčević.

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