On the polynomial numerical hull of a normal matrix.
On upper and lower bounds of the numerical radius and an equality condition
We give an inequality relating the operator norm of T and the numerical radii of T and its Aluthge transform. It is a more precise estimate of the numerical radius than Kittaneh's result [Studia Math. 158 (2003)]. Then we obtain an equivalent condition for the numerical radius to be equal to half the operator norm.
Orbits in strips
Product of operators and numerical range preserving maps
Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of by . This includes the usual product and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying such that ϕ: V → V has the form A...
Properties of lush spaces and applications to Banach spaces with numerical index 1
The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably...
Quasinormality and numerical ranges of certain classes of dual Toeplitz operators.
Recent progress and open questions on the numerical index of Banach spaces.
The aim of this paper is to review the state-of-the-art of recent research concerning the numerical index of Banach spaces, by presenting some of the results found in the last years and proposing a number of related open problems.
Reflexive spaces and numerical radius attaining operators.
In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].
Several variable spectral theory in the noncommutative case
Shapes and computer generation of numerical ranges of Krein space operators.
Some Berezin number inequalities for operator matrices
Some properties of composition operators on the Dirichlet space.
Spatial numerical range of operators on weighted Hardy spaces.
Spectra and Numerical Ranges in Locally M-Convex Algebras
Spectral theory and Cauchy problems on Lp spaces.
Spectraloid operator polynomials, the approximate numerical range and an Eneström-Kakeya theorem in Hilbert space
We study a class of operator polynomials in Hilbert space which are spectraloid in the sense that spectral radius and numerical radius coincide. The focus is on the spectrum in the boundary of the numerical range. As an application, the Eneström-Kakeya-Hurwitz theorem on zeros of real polynomials is generalized to Hilbert space.
Sums of idempotents and a lemma of N. J. Kalton
A lemma of Gelfand-Hille type is proved. It is used to give an improved version of a result of Kalton on sums of idempotents.
The Bishop-Phelps-Bollobás property for numerical radius in ℓ₁(ℂ)
We show that the set of bounded linear operators from X to X admits a Bishop-Phelps-Bollobás type theorem for numerical radius whenever X is ℓ₁(ℂ) or c₀(ℂ). As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollobás theorem for ℓ₁(ℂ).
The hypo-Euclidean norm of an -tuple of vectors in inner product spaces and applications.
The joint essential numerical range, compact perturbations, and the Olsen problem
Let T₁,...,Tₙ be bounded linear operators on a complex Hilbert space H. Then there are compact operators K₁,...,Kₙ ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T₁-K₁,...,Tₙ-Kₙ) equals the joint essential numerical range of (T₁,...,Tₙ). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ ℕ then there exists a compact operator K ∈ B(H) such that . This generalizes results of C. L. Olsen.