Advances in Operator Theory最新文献

In the present paper we introduce a method for generating ideals of linear and multilinear operators from what we call generalized left and right operator ideals, that we discuss with p-th power factorable, p-convex and q-concave operators. Then we combine this method with the Factorization Ideal method, that construct multilinear operators, in order to introduce the ideal of multilinear ({mathcal {F}}_{vec {p},vec {q}})-factorable operators as an example of an ideal generated by means of our method. Finally, we investigate its relation with multilinear summing operators.

Let (mathfrak {M}) be a von Neumann algebra. For a nonzero positive element (Ain mathfrak {M}), let P denote the orthogonal projection on the norm closure of the range of A and let (sigma _A(T) ) denote the A-spectrum of any (Tin mathfrak {M}^A). In this paper, we show that (sigma _A(T)) is a non empty compact subset of (mathbb {C}) and that (sigma (PTP, Pmathfrak {M}P)subseteq sigma _A(T)) for any (Tin mathfrak {M}^A) where (sigma (PTP, Pmathfrak {M}P)) is the spectrum of PTP in (Pmathfrak {M}P). Sufficient conditions for the equality (sigma _A(T)=sigma (PTP, Pmathfrak {M}P)) to be true are also presented. Moreover, we show that (sigma _A(T)) is finite for any (Tin mathfrak {M}^A) if and only if A is in the socle of (mathfrak {M}). Furthermore, we consider the relationship between elements S and (Tin mathfrak {M}^A) that satisfy the condition (sigma _A(SX)=sigma _A(TX)) for all (Xin mathfrak {M}^A). Finally, a Gleason–Kahane–Żelazko’s theorem for the A-spectrum is derived.

We say that a (C^*)-algebra ({mathcal {A}}) satisfies the similarity property ((SP)) if every bounded homomorphism (u: {mathcal {A}}rightarrow {mathcal {B}}(H),) where H is a Hilbert space, is similar to a (*)-homomorphism and that a von Neumann algebra ({mathcal {M}}) satisfies the weak similarity property ((WSP)) if every (textrm{w}^*)-continuous, unital and bounded homomorphism (pi : {mathcal {M}}rightarrow {mathcal {B}}(H),) where H is a Hilbert space, is similar to a (*)-homomorphism. The similarity problem is known to be equivalent to the question of whether every von Neumann algebra is hyperreflexive. We improve on that by introducing the following hypothesis (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive. We prove that under (EP), every separably acting von Neumann algebra satisfies (WSP) and we pass from the case of separably acting von Neumann algebras to all (C^*)-algebras.

Pólya-type functions are of special importance in probability and harmonic analysis. We introduce and study their multidimensional extensions.

Let A, B be any two positive definite (ntimes n) matrices and Y be any (ntimes n) matrix. The matrices (M_Y(A,B)=left[ begin{array}{cc} A &{} A^{frac{1}{2}}YB^{frac{1}{2}} B^{frac{1}{2}}Y^{star }A^{frac{1}{2}} &{} B end{array}right] ) for Y to be contractive, expansive or unitary matrix, are in fact arising from matrix/operator means. We aim to establish the signatures of the eigenvalues of the sum of two matrices of the type (M_Y(A,B).) We characterise any (ntimes n) matrix A through its dilations given by ({mathcal {P}}_3(A)=begin{bmatrix} O &{} A &{} A^2 A^* &{} O &{} A {A^*}^2 &{} A^* &{} O end{bmatrix}) and ({mathcal {M}}_3(A)=begin{bmatrix} I &{} A &{} A^2 A^* &{} I &{} A {A^*}^2 &{} A^* &{} I end{bmatrix},) by means of inertia of dilations.

Isometric covariant representations play an important role in the study of Cuntz–Pimsner algebras. In this article, we study partial isometric covariant representations and explore under what conditions powers and roots of partial isometric covariant representations are also partial isometric covariant representations.

Let X be a real or complex Banach space of dimension at least 3. We give a complete description of surjective mappings on B(X) that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators.

Under an abstract setting, we show that eigenvectors belong to discrete spectra of unitary operators have exponential decay properties. We apply the main theorem to multi-dimensional quantum walks and show that eigenfunctions belong to a discrete spectrum decay exponentially at infinity.

In this paper, we are starting to construct a new theory of absolutely simple p-summing operators. We define a significant class of weak operator ideals, namely the class of absolutely simple p-summing operators between arbitrary real Banach spaces and show some basic properties of that class. A key feature of the resulting class is computing simple p-summing norms exactly for any linear operator between finite-dimensional normed spaces, in contrast to the computation of p-summing norms which is in general difficulty or the computation of Lipschitz p-summing norms between particular classes of metric spaces. Building upon S. Kwapień’s result, we figure out the relations between 2-summing norms and simple 2-summing norms and find out the relations between simple p-summing norms and diverse familiar norms of some linear operators. In the end, we present some concluding remarks and introduce some open problems that we think are intriguing.