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In this paper, we introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of Doperators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the question of when the equations A ⁎ s A s = ( A ⁎ A ) s , s ∈ S , where S is a finite set of positive integers, imply the quasinormality or normality of A . In particular, it is proved that if S = { p , m , m + p , n , n + p } , where p ⩾ 1 and 2 ⩽ m < n , then A is quasinormal. Moreover, if A is invertible and S = { m , n , n + m } with m ⩽ n , then A is normal. The case when S = { m , m + n } and A ⁎ n A n ⩽ ( A ⁎ A ) n is also discussed. [ABSTRACT FROM AUTHOR]

In this paper we find a formula for the adjoint of bounded pseudo-differential operators on compact Lie groups, which yields a necessary and sufficient condition for self-adjointness of these operators. By a similar method we find necessary and sufficient conditions for bounded pseudo-differential operators on compact Lie groups to be normal or unitary and we give some examples of our results on . Then we show that similar results are true for some bounded pseudo-differential operators on . [ABSTRACT FROM AUTHOR]

In this paper we characterize self-adjoint and normal composition operators on Poisson weighted sequence spaces l²(λ). However, the main purpose of this paper is to determine explicit conditions on inducing map under which a composition operator admits a best normal approximation. We extend results of Tripathi and Lal [Antinormal composition operators on l², Tamkang J. Math. 39 (2008), 347-352] to characterize antinormal composition operators on l²(λ). [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]

In this paper, we study the Fuglede-Putnam property. We give a necessary and sufficient condition for which $$(\oplus _{i=1}^{n}A_i,\oplus _{i=1}^{n}B_i)$$ satisfies the Fuglede-Putnam property. We also study the local spectral theory associated with the Fuglede-Putnam property. Finally, we define the weak Fuglede-Putnam property and we investigate several cases which satisfy the weak Fuglede-Putnam property. [ABSTRACT FROM AUTHOR]

The matrix operator defined by the left-hand side of the Bevis–Hall–Hartwig equation(briefly, the BHH operator) is not linear in the space of complex matrices. However, it becomes a linear operator if every matrixis replaced by the pair of real matrices of the same size, namely, by its real and imaginary parts. Introducing an appropriate scalar product in the resulting real space, we find necessary and sufficient conditions for the BHH operator to be normal. [ABSTRACT FROM PUBLISHER]

An operator T is called n-normal operator if TnT* = T*Tn and n-quasinormal operator if TnT*T = T*TTn. In this paper, the conditions under which composition operators and weighted composition operators become n-normal operators and n-quasinormal operators have been obtained in terms of Radon-Nikodym derivative hn. [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]

The concept of maximal quasimonotonicity of set-valued map is introduced and studied. Regularity properties of this class of operators is investigated, in particular the ccvc property, an adaptation to the quasimonotone case of the classical notion of cusco map. In an Asplund space, we provide sufficient conditions for a ccvc quasimonotone operator to be single-directional on a $$G_\delta $$ -dense subset. [ABSTRACT FROM AUTHOR]

Let A = U| A| and B = V| B| be the polar decompositions of $${A \in \mathbb{B}(\fancyscript{H}_1)}$$ and $${B\in \mathbb{B}(\fancyscript{H}_2)}$$ and let Com( A, B) stand for the set of operators $${X \in \mathbb{B}(\fancyscript{H}_2,\fancyscript{H}_1)}$$ such that AX = XB. A pair ( A, B) is said to have the FP-property if Com( A, B) $${\subseteq}$$ Com( A*, B*). Let $${\tilde{C}}$$ denote the Aluthge transform of a bounded linear operator C. We show that (i) if A and B are invertible and ( A, B) has the FP-property, then so is $${(\tilde{A}, \tilde{B})}$$ ; (ii) if A and B are invertible, the spectrums of both U and V are contained in some open semicircle and $${(\tilde{A}, \tilde{B})}$$ has the FP-property, then so is ( A, B); (iii) if ( A, B) has the FP-property, then Com( A, B) $${\subseteq}$$ Com $${(\tilde{A},\tilde{B})}$$, moreover, if A is invertible, then Com( A, B) = Com $${(\tilde{A}, \tilde{B})}$$. Finally, if $${\mbox{Re}(U|A|^{1\over2})\geq a>0}$$ and $${\mbox{Re}(V|B|^{1\over2})\geq a>0}$$ and X is an operator such that U* X = XV, then we prove that $${\|\tilde{A}^* X-X\tilde{B}\|_p\geq 2a\|\,|B|^{1\over2}X-X|B|^{1\over2} \|_p}$$ for any $${1 \leq p \leq \infty}$$. [ABSTRACT FROM AUTHOR]

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: 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]

Dontchev and Hager [Math. Oper. Res., 19 (1994), pp. 753-768] have shown that a monotone set-valued map defined from a Banach space to its dual which satisfies the Aubin property around a point (x, y) of its graph is actually single-valued in a neighborhood of x. We prove a result which is the counterpart of the above for quasi-monotone set-valued maps, based on the concept of single-directional property. As applications, we provide sufficient conditions for this single-valued property to hold for the solution map of general variational systems and quasi-variational inequalities. We also investigate the single-directionality property for the normal operator to the sublevel sets of a quasi-convex function. [ABSTRACT FROM AUTHOR]

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 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]

We establish sufficient conditions for the solvability of a fourth order operator differential equation. These conditions are expressed in terms of operator coefficients. [ABSTRACT FROM AUTHOR]

In this article we prove an existence result, necessary and sufficient conditions for quasiconvex programming problem with a locally starshaped constraint region. Our optimality conditions are different from the usual optimality conditions, in that the subdifferential of the objective function is replaced by a normal cone operator. Such an optimality condition has advantage over the usual one, i.e. it becomes sufficient even when the objective function is only quasiconvex. As a special case we derive the corresponding results for the class of ‘Quasiconvex-quasiaffine’ MPEC which is a class of mathematical programs with complementarity constraints where the objective function is quasiconvex, the inequality constraint is quasiconvex and the rest of constraints are quasiaffine. [ABSTRACT FROM AUTHOR]

In this note, we present three simple necessary and sufficient conditions for a linear compact operator on a Hilbert space H to be normal. [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]