Written by a distinguished specialist in functional analysis, this book presents a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory ...is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference for researchers and as an introduction for graduate students who want to learn Banach space theory with some historical flavor.
We study the isomorphic structure of
$(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$
and prove that these spaces are complementably homogeneous. We also show that for any operator T from
$(\sum {\ell ...}_{q})_{c_{0}}$
into
${\ell }_{q}$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
that is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
and the restriction of T on X has small norm. If T is a bounded linear operator on
$(\sum {\ell }_{q})_{c_{0}}$
which is
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular, then for any
$\epsilon>0$
, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$
which is isometric to
$(\sum {\ell }_{q})_{c_{0}}$
with
$\|T|_{X}\|<\epsilon $
. As an application, we show that the set of all
$(\sum {\ell }_{q})_{c_{0}}$
-strictly singular operators on
$(\sum {\ell }_{q})_{c_{0}}$
forms the unique maximal ideal of
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$
.
Let A be a bounded linear operator on a complex Hilbert space and ℜ(A) (ℑ(A)) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we ...prove thatw(A)≥12‖A‖+12|‖ℜ(A)‖−‖ℑ(A)‖|andw2(A)≥14‖A⁎A+AA⁎‖+12|‖ℜ(A)‖2−‖ℑ(A)‖2|, where w(A) and ‖A‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for w(A)=12‖A⁎A+AA⁎‖ and prove that w(A)=12‖A⁎A+AA⁎‖ if and only if the numerical range of A is a circular disk with center at the origin and radius 12‖A⁎A+AA⁎‖. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.
Abstract
The goal of this paper, is to introduce another classes of the fuzzy soft bounded linear operator in the fuzzy soft Hilbert space which is a fuzzy soft quasi normal operator, as well as, ...give some properties about this concept with investigating the relationship among this types of the fuzzy soft bounded linear operator on fuzzy soft Hilbert space with other kinds of fuzzy soft bounded linear operators.
The paper is concerned with the study of the limit behaviour of the sequences of the positive linear functionals and operators associated with integrated generalized means defined with respect to a ...given probability Borel measure in the framework of Borel convex subsets of a Hilbert space. The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space. Several applications are shown in the special cases of bounded and unbounded real intervals which involve the most common integrated means. Furthermore, some consequences concerning the convergence in distribution, and hence stochastic, of generalized means of vector-valued random variables are also presented. Finally the paper ends with an application related to the so-called box integral problem which refers to the problem to evaluate the limit behaviour as n→∞ of the average distance between two points of 0,1n randomly chosen according to a given distribution on 0,1n.
We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present ...an upper bound for the spectral radius of sum of product of n pairs of operators. As an application of the results obtained, we provide a better estimation for the zeros of a given polynomial.
We consider the problem of estimating functionals of discrete distributions, and focus on a tight (up to universal multiplicative constants for each specific functional) nonasymptotic analysis of the ...worst case squared error risk of widely used estimators. We apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional, namely their bias. We explicitly characterize the worst case squared error risk incurred by the maximum likelihood estimator (MLE) in estimating the Shannon entropy H(P) = Σ i=1 S -p i ln p i , and the power sum F α (P) = Σ i=1 S p i α , α > 0, up to universal multiplicative constants for each fixed functional, for any alphabet size S ≤ ∞ and sample size n for which the risk may vanish. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have n ≫ S observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider n ≫ S 1/α samples for the MLE to consistently estimate F α (P), 0 <; α <; 1. The minimax rate-optimal estimators for both problems require S/ ln S and S 1/α / ln S samples, which implies that the MLE has a strictly sub-optimal sample complexity. When 1 <; α <; 3/2, we show that the worst case squared error rate of convergence for the MLE is n -2(α-1) for infinite alphabet size, while the minimax squared error rate is (n ln n) -2(α-1) . When α ≥ 3/2, the MLE achieves the minimax optimal rate n -1 regardless of the alphabet size. As an application of the general theory, we analyze the Dirichlet prior smoothing techniques for Shannon entropy estimation. In this context, one approach is to plug-in the Dirichlet prior smoothed distribution into the entropy functional, while the other one is to calculate the Bayes estimator for entropy under the Dirichlet prior for squared error, which is the conditional expectation. We show that in general such estimators do not improve over the maximum likelihood estimator. No matter how we tune the parameters in the Dirichlet prior, this approach cannot achieve the minimax rates in entropy estimation. The performance of the minimax rate-optimal estimator with n samples is essentially at least as good as that of Dirichlet smoothed entropy estimators with n ln n samples.
If
A
,
B
are bounded linear operators on a complex Hilbert space, then we prove that
w
(
A
)
≤
1
2
‖
A
‖
+
r
|
A
|
|
A
∗
|
,
w
(
A
B
±
B
A
)
≤
2
2
‖
B
‖
w
2
(
A
)
-
c
2
(
R
(
A
)
)
+
c
2
(
I
(
A
)
)
...2
,
where
w
(
·
)
,
·
, and
r
(
·
)
are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and
R
(
A
)
,
I
(
A
)
are the real part, the imaginary part of
A
respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.
This study discussed the integral of Dunford and compact linear operator on space of Dunford integral function. For each f which is Dunford integral on a,b is defined as an operator DL by DL (x*) = ...x*f, for each x* ∈ X* This study resulted that the operator DL is both a continuous linear operator and weakly compact operators. Then, it was defined as the adjoint of the operator DL* by DL*(h)(x*)=∫abhDL(x*) each h ∈ (L1)* The adjoint operator DL* is continuous and weakly compact linear operators.
This paper is concerned with the relatively bounded perturbations of a closed linear relation and its adjoint in Hilbert spaces. A stability result about orthogonal projections onto the ranges of ...linear relations is obtained. By using this result, two perturbation theorems for a closed relation and its adjoint are given. These results generalize the corresponding results for single-valued linear operators to linear relations and some of which weaken certain assumptions of the related existing results.
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