A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

Property (quasi‐α) and the denseness of norm attaining mappings

We introduce property (quasi ‐α), which implies property (A) defined by Lindenstrauss [10] and whose dual property is property (quasi‐β) [2]. We consider relations between this property and other sufficient conditions for property (A), and study the denseness of norm attaining mappings under the conditions of these properties. In particular, if each of the Banach spaces X_k , 1 ≤ k ≤ n – 1, has property (quasi‐α) and X_n has property (A), then the projective tensor product X₁ ··· X_n has property (A). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.     

A uniform boundedness theorem for locally convex cones

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

     

Landau’s theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk D$\mathbb{D}$. Then, we obtain two versions of Landau’s theorem for polyharmonic mappings F$F$, and for the mappings of the type L(F)$L\left(F\right)$, where L$L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.     

Polaroid operators and Weyl's theorem

``Polaroid elements" represent an attempt to abstract part of the condition, ``Weyl's theorem holds" for operators.

     

Analytically Class A operators and Weyl's theorem

Using the new spectrum set defined in this note, we give the necessary and sufficient condition for T which the Weyl's theorem holds. We also consider how the Weyl's theorem survives for analytically Class A operators.     

Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

Weyl type theorem and hypercyclic operators

Using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for T for which the a-Browder's theorem or the a-Weyl's theorem holds. Also, the relation between hypercyclic operators (or supercyclic operators) and the operators which satisfy Weyl type theorem is discussed.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

A Hörmander type multiplier theorem for multilinear operators

In this paper, we prove a Hörmander type multiplier theorem for multilinear operators. As a corollary, we can weaken the regularity assumption for multilinear Fourier multipliers to assure the boundedness.     

Weyl's theorem for perturbations of paranormal operators

A bounded linear operator on a Banach space is said to satisfy ``Weyl's theorem' if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if is a paranormal operator on a Hilbert space, then satisfies Weyl's theorem for every algebraic operator which commutes with .

     

Weyl's theorem holds for algebraically hyponormal operators

In this note it is shown that if is an ``algebraically hyponormal" operator, i.e., is hyponormal for some nonconstant complex polynomial , then for every , Weyl's theorem holds for , where denotes the set of analytic functions on an open neighborhood of .

     

A uniqueness theorem for indefinite Sturm-Liouville operators

In this paper, we discuss the inverse problem for indefinite Sturm-Liouville operators on the finite interval [a, b]. For a fixed index n(n = 0, 1, 2, ··· ), given the weight function ω(x), we will show that the spectral sets {λ n (q, h a , h k )} +∞ k=1 and {λ-n (q, h b , h k )} +∞ k=1 for distinct h k are sufficient to determine the potential q(x) on the finite interval [a, b] and coefficients h a and h b of the boundary conditions.     

A lifting theorem for unbounded quasinormal operators

Relationships between minimal normal extensions of spectral and cyclic type of an unbounded quasinormal operator are discussed and some properties such as, for example, tightness of such extensions are established. A Yoshino type criterion on the lifting of the strong commutant of an unbounded quasinormal operator is proved.     

Lomonosov's invariant subspace theorem for multivalued linear operators

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator on an infinite-dimensional normed space ``commutes' with a compact operator i.e., then has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide an application to single-valued linear operators.