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Anderson's theorem states that if the numerical range W ( A ) of an n -by- n matrix A is contained in the unit disk D ‾ and intersects with the unit circle at more than n points, then W ( A ) = D ‾ . An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the Hardy space H 2 . Moreover, if T φ is a complex symmetric Toeplitz operator, we provide a necessary and sufficient condition for T φ to be normal. Finally, we investigate these T φ with finite symbols. [ABSTRACT FROM AUTHOR]

We establish an algorithm for a criterion of the diagonalisability of a matrix over a local field by a unitary matrix. For this sake, we define the notion of normality of a p -adic operator, and give several criteria for the normality. We study the relation between the normality and the reduction. In the finite dimensional case, the normality of an operator is equivalent to the diagonalisability of a matrix by a unitary matrix. Therefore we also study the relation between the diagonalisability and the reduction. For example, we show that the diagonalisation of the reduction gives a partition of unity corresponding to the reduction of the spectrum, which gives a functorial lift of the eigenspace decomposition of the reduction. [ABSTRACT FROM AUTHOR]

A bounded operator T on a separable Hilbert space H is said to be complex symmetric if there exists an orthonormal basis for H with respect to which T has a self-transpose matrix representation. In this paper, we study the complex symmetry of composition operators C ϕ f = f ∘ ϕ induced on the Hardy space H 2 by holomorphic self-maps ϕ of the open unit disk D . For any holomorphic self-map ϕ of D , we establish that if C ϕ is complex symmetric, then ϕ must fix a point in D . Thus among the automorphisms of D , only the elliptic ones may induce complex symmetric composition operators. For an elliptic automorphism ϕ , we prove that if ϕ is not a rotation or of order 3, then C ϕ is complex symmetric if and only if ϕ ( z ) = α − z 1 − α ¯ z for some α ∈ D ∖ { 0 } . [ABSTRACT FROM AUTHOR]

Abstract: We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite -algebras. A simple proof to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial -group will be given using previously known techniques. Using these techniques some upper and lower bounds for the distance between unitary orbits of normal operators will be obtained. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infinite -algebras and type III factors with separable predual. [Copyright &y& Elsevier]

Abstract: We introduce the spectral points of two-sided positive type of bounded normal operators in Krein spaces. It is shown that a normal operator has a local spectral function on sets which are of two-sided positive type. In addition, we prove that the Riesz–Dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive. The restriction of the operator to this subspace is then normal in a Hilbert space. [Copyright &y& Elsevier]

Abstract: A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for H-selfadjointness of an H-symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples. [Copyright &y& Elsevier]

Abstract: This paper is a continuation of the study by Foias, Jung, Ko, and Pearcy (2007) and Foias, Jung, Ko, and Pearcy (2008) of rank-one perturbations of diagonalizable normal operators. In Foias, Jung, Ko, and Pearcy (2007) we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace, and in Foias, Jung, Ko, and Pearcy (2008) we proved that the commutant of each of these rank-one perturbations is abelian. In this paper we show that the operators considered in Foias, Jung, Ko, and Pearcy (2007) have more structure – namely, that they are decomposable operators in the sense of Colojoară and Foias (1968) . [Copyright &y& Elsevier]

Abstract: Let φ be an analytic function on the open unit disc such that , and let ψ be an analytic function on such that the weighted composition operator defined by is bounded on the Hardy space . We characterize those weighted composition operators on that are unitary, showing that in contrast to the unweighted case (), every automorphism of induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators on for which the inducing map φ fixes a point in . This description shows both ψ and φ must be linear fractional in order for to be normal (assuming φ fixes a point in ). In general, we show that if is normal on and , then φ must be either univalent on or constant. Descriptions of spectra are provided for the operator when it is unitary or when it is normal and φ fixes a point in . [Copyright &y& Elsevier]

Abstract: Let be the -algebra of all bounded linear operators acting on a complex Hilbert space H. In this note, we shall show that if S is an invertible normal operator in the following estimation holds where is the injective norm on the tensor product . This last inequality becomes an equality when S is invertible self-adjoint. On the other hand, we shall characterize the set of all invertible normal operators S in satisfying the relation and also we shall give some characterizations of some subclasses of normal operators in by inequalities or equalities. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H where is a positive definite self-adjoint operator, and satisfy Caratheodory condition. We present some spectral conditions for the nonlinearity to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of the self-adjoint elliptic boundary value problems. [Copyright &y& Elsevier]

In the article [8] , an example of a composition operator C φ on the Hardy space H 2 ( D ) is discovered that is claimed to be complex symmetric but not normal. The purpose of this note is to highlight a mistake in the example that makes the result incorrect. [ABSTRACT FROM AUTHOR]

Abstract: The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed. [Copyright &y& Elsevier]

Abstract: Friedland''s characterization of bounded normal operators is shown to hold for infinitesimal generators of -semigroups. New criteria for normality of bounded operators are furnished in terms of Hamburger moment problem. All this is achieved with the help of the celebrated Ando''s theorem on paranormal operators. [Copyright &y& Elsevier]

Abstract: It is proved that any finite-dimensional solvable Lie algebra generated by normal operators is commutative, and some other related results are given. [Copyright &y& Elsevier]

Abstract: This paper is concerned with operators on Hilbert space of the form where D is a diagonalizable normal operator and is a rank-one operator. It is shown that if and the vectors u and v have Fourier coefficients and with respect to an orthonormal basis that diagonalizes D that satisfy , then T has a nontrivial hyperinvariant subspace. This partially answers an open question of at least 30 years duration. [Copyright &y& Elsevier]

Abstract: Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standard rigged Hilbert space constructions and avoids the introduction of nuclear spaces. The techniques are predominantly measure theoretic and hence the Hilbert spaces involved are separable. The results range from a Naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators. [Copyright &y& Elsevier]

Abstract: It is proved that the operator Lie algebra generated by a bounded linear operator T on Hilbert space is finite-dimensional if and only if , N is a normal operator, , and , where denotes the smallest Lie algebra containing , and denotes the associative subalgebra of generated by . Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if is an ad-compact E-solvable Lie algebra, then T is a normal operator. [Copyright &y& Elsevier]

It is shown that if A, B, X are Hilbert space operators such that X⩾γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then &z.sfnc;A−B&z.sfnc;2⩽p&z.sfnc;A&z.sfnc;2+q&z.sfnc;B&z.sfnc;2 with equality if and only if (1−p)A=B and γ&z.sfnc;&z.sfnc;&z.sfnc;&z.sfnc;A−B&z.sfnc;2&z.sfnc;&z.sfnc;&z.sfnc;⩽&z.sfnc;&z.sfnc;&z.sfnc;p&z.sfnc;A&z.sfnc;2X+qX&z.sfnc;B&z.sfnc;2&z.sfnc;&z.sfnc;&z.sfnc; for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert–Schmidt operator, then &z.dfnc;δA,B2(X)&z.dfnc;2⩽&z.dfnc;p&z.sfnc;A&z.sfnc;2X+qX&z.sfnc;B&z.sfnc;2&z.dfnc;2 with equality if and only if (1−p)AX=XB. [Copyright &y& Elsevier]

Two types of commutator inequalities for the Hilbert–Schmidt norm are established. The first type of these inequalities is related to a classical inequality of Clarkson, and the second type is related to the unitary approximation of positive and invertible operators. [Copyright &y& Elsevier]