Characterizations of Hilbert space and the Vidav-Palmer theorem (original) (raw)
Related papers
2017
Let be a Hilbert space equipped with the inner product , and let be the algebra of bounded linear operators acting on . We recall that the numerical range (also known as the field of values) of is the collection of all complex numbers of the form where is a unit vector in . i.e. See, ([2], [5], [8]) which is useful for studying operators. In particular, the geometrical properties of the numerical range often provide useful information about the algebraic and analytic properties of the operator . For instance, if and only if ; is real if and only if , has no interior points if and only if there are complex numbers, and with such that is self-adjoint. Moreover, the closure of denoted by , always contains the spectrum of denoted by . See, [8] Let denote the set of compact operators on and be the canonical quotient map. The essential numerical range of , denoted by is the set; See, ([1], [2], [3]) where the intersection runs over the compact operators . Chacon and Chacon [3] gave some o...
Numerical range and orthogonality in normed spaces
Filomat, 2009
Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V (T)A which generalize those given by J.P. William and J.P. Stamplfli. Finally, we give a positive answer of the Mathieu's question. .
On operators preserving the numerical range
Linear Algebra and its Applications, 1990
Let F be a surjective linear mapping between the algebras L(H) and L(K) of all bounded operators on nontrivial complex Hilbert spaces H and K respectively. For any positive integer k let W,(A) denote the kth numerical range of an operator A on H. If k is strictly less than one-half the dimension of H and W,(F(A)) = Wk. A) for ah A from L(H), then there is a unitary mapping U: H + K such that either F(A) = UAu* or F(A) = (UAU*)' for every A E L(H), where the transposition is taken in any basis of K, fixed in advance. This generalizes the result of S. Pierce and W. Watkins on finite-dimensional spaces. The case of k greater than or equal to one-half of the dimension of H is also treated using our method. Our proofs depend on a characterization of those linear operators preserving projections of rank one, which is of independent interest.
On the Numerical Range and Norm of Elementary Operators
Linear and Multilinear Algebra, 2004
W 0 ðR A, B J j Þ where VðÁÞ is the joint spatial numerical range, W 0 ðÁÞ is the algebraic numerical range and J is a norm ideal of BðEÞ: We shall show that this inclusion becomes an equality when R A, B is taken to be a derivation. Also, we deduce that wðU A, B J j Þ ! 2ð ffiffi ffi 2 p À 1ÞwðAÞwðBÞ, for A, B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÁÞ is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound kU A, B jJk ! 2ð ffiffi ffi 2 p À 1ÞkAkkBk holds, if one of the following two conditions is satisfied: (i) J is a standard operator algebra of BðEÞ and A, B 2 J: (ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:
Hilbert Operator Spaces with Applications
2015
The purpose of this paper is to Provide conditions for the existence of farthest points of closed and bounded subsets of Hilbert operator spaces. This will done by applying the concept of numerical range. We give, inter alia, some results to characterize farthest points of a subset of a C *-algebra A from a fixed element x ∈ A. Meanwhile, we point out the main theorems of R. Saravanan and R. Vijayaragavan[11] are incorrect, by given two counterexamples.
Editors’ foreword for the special issue “Mathematics in the Banach Space”
European Journal of Mathematics
The notion of a Banach space is one of the most fundamental notions of modern mathematics. Such spaces were named to honour Stefan Banach (1892-1945), one of the founders of Functional Analysis, who lived, worked and died in Lviv (now the largest city in western part of Ukraine). Of course, there are many important Banach spaces: spaces of sequences, functions, operators, etc. Yet, there exists one very concrete Banach space, called the Banach space. It includes numerous historical places in Lviv related to Stefan Banach: the houses where B Taras Banakh
On Bounded Linear Operations in b-Hilbert Spaces and their Numerical Ranges
2015
In this paper, we introduce the notions of b-bounded linear operator, b-numericalrange and b-numerical radius in a b-Hilbert space and describe some of their properties. Thenwe will show that this new numerical range (radius) can be considered as a usual numericalrange (radius) in a Hilbert space, so it shares many useful properties with numerical range(radius).