The numerical range as a spectral set (original) (raw)
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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.
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...
Constructive spectral and numerical range theory
2011
Following an introductory chapter on constructive mathematics, Chapter 2 contains a detailed constructive analysis of the Toeplitz-Hausdorff Theorem on the convexity of the numerical range of an operator in a Hilbert space. It is shown that the results in the chapter are the best possible with constructive methods. The rest of the thesis deals with the constructive theory of not-necessarily-commutative Banach algebras. Chapter 3 discusses the Spectral Mapping Theorem in that context, again showing that the results obtained are the best possible. Chapter 4 deals with the question, "Are positive integral powers of a hermitian element of a Banach algebra hermitian?" A major problem that has to be overcome is to find the 'right' constructive definition of hermitian, since there is no guarantee in constructive mathematics that the state space of a Banach algebra is nonempty; this forces us to work with approximations to the state space, rather than the state space itsel...
A note on the maximal numerical range
Operators and Matrices, 2019
We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given. First, let us set some notation and terminology. For a subset X of the complex plane C, by cl X, ∂X, and conv X we will denote the closure, boundary, and the convex hull of X, respectively. By an "operator" we throughout the paper understand a bounded linear operator acting on a Hilbert space H. The numerical range of such an operator A is defined by the formula W (A) = { Ax, x : x ∈ H, x = 1}, where .,. and. stand, respectively, for the scalar product on H and the norm associated with it. Introduced a century ago in the works by Toeplitz [8] and Hausdorrf [6] (and thus also known as the Toeplitz-Hausdorff set), it since has been a subject of intensive research. We mention here only [4] as a standard source of references, and note the following basic properties: Due to the Cauchy-Schwarz inequality, the set W (A) is bounded. Namely, w(A) := sup{|z| : z ∈ W (A)} ≤ A ; (1) w(A) is called the numerical radius of A.
On Generalized Numerical Ranges of Operators on an Indefinite Inner Product Space
Linear and Multilinear Algebra, 2004
In this paper, numerical ranges associated to operators on an indefinite inner product space are investigated. Boundary generating curves, corners, shapes and computer generations of these sets are studied. In particular, the Murnaghan-Kippenhahn theorem for the classical numerical range is generalized.
Numerical ranges of an operator on an indefinite inner product space
The Electronic Journal of Linear Algebra, 1996
For n n complex matrices A and an n n Hermitian matrix S, w e consider the S-numerical range of A and the positive S-numerical range of A de ned by S A. Possible generalizations of our results, including their extensions to bounded linear operators on an in nite dimensional Hilbert or Krein space, are discussed.
The numerical range of elementary operators II
Linear Algebra and its Applications, 2001
For A, B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H), it is proved that: (i) the generalized derivation δ A,B is convexoid if and only if A and B are convexoid; (ii) the operators δ A,B and δ A,B |C p (where p 1) have the same numerical range and are equal to W 0 (A) − W 0 (B) (where C p is the Banach space of the p-Schatten class operators on H).