Advances in Operator Theory最新文献

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A,  a classical proposal for the weight vector, and the Perron eigenvector of (AA^{T}) (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.

We show that the c-numerical range of a non-scalar skew-Hermitian quaternion matrix is convex. In fact, included in our result is that the c-numerical range of a skew-Hermitian matrix is a rotation invariant subset of the quaternions with zero real parts.

In this paper, we generalize and investigate the concept of Cesàro hypercyclicity of linear operators for linear relations. In addition, we provide new characterizations and properties for this concept.

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator (T^mu _tau ),

in suitable special Besov spaces (B^s_p ({{mathbb {R}}}^n) = B^s_{p,p} ({{mathbb {R}}}^n)), (s>0), (1<p<infty ). Here (tau (x,xi )) are the symbols of (smooth) pseudodifferential operators belonging to appropriate Hörmander classes (Psi ^sigma _{1, delta } ({{mathbb {R}}}^n)), (sigma <0), (0 le delta le 1) (including the exotic case (delta =1)) whereas (mu ) is the Hausdorff measure of a compact d–set (Gamma ) in ({{mathbb {R}}}^n), (0<d<n). This extends previous assertions for the positive-definite selfadjoint fractal differential operator ((textrm{id}- Delta )^{sigma /2} mu ) based on Hilbert space arguments in the context of suitable Sobolev spaces (H^s ({{mathbb {R}}}^n) = B^s_2 ({{mathbb {R}}}^n)). We collect the outcome in the Main Theorem below. Proofs are based on estimates for the entropy numbers of the compact trace operator

We add at the end of the paper a few personal reminiscences illuminating the role of Pietsch in connection with the creation of approximation numbers and entropy numbers.

Let (mathcal {M}) be a von Neumann algebra of operators on a Hilbert space (mathcal {H}) and (tau ) be a faithful normal semifinite trace on (mathcal {M}), (S(mathcal {M}, tau )) be the ( ^*)-algebra of all (tau )-measurable operators. Assume that an operator (Tin S(mathcal {M}, tau )) is paranormal or ( ^*)-paranormal. If (T^n) is (tau )-compact for some (nin mathbb {N}) then T is (tau )-compact; if (T^n=0) for some (nin mathbb {N}) then (T=0); if (T^3=T) then (T=T^*); if (T^2in L_1(mathcal {M}, tau )) then (Tin L_2(mathcal {M}, tau )) and (Vert TVert _2^2=Vert T^2Vert _1). If an operator (Tin S(mathcal {M}, tau )) is hyponormal and (T^{*p}T^q) is (tau )-compact for some (p, q in mathbb {N}cup {0}), (p+q ge 1) then T is normal. If (Tin S(mathcal {M}, tau )) is p-hyponormal for some (0<ple 1) then the operator ((T^*T)^p-(TT^*)^p) cannot have the inverse in ( mathcal {M}). If an operator (Tin S(mathcal {M}, tau )) is hyponormal (or cohyponormal) and the operator (T^2) is Hermitian then T is normal.

The method of cyclic resolvents has been extended for a finite family of quasi-convex functions and quasi-nonexpansive mappings in Hadamard spaces. The essential tool for proving the main results is the use of the recent article by the first author and Mohebbi on the behavior of an iteration of a strongly quasi-nonexpansive sequence. The results are new even in Hilbert spaces.

Singular numbers of linear operators between Hilbert spaces were generalized to Banach spaces by s-numbers (in the sense of Pietsch). This allows for different choices, including approximation, Gelfand, Kolmogorov and Bernstein numbers. Here, we present an elementary proof of a bound between the smallest and the largest s-number.

Determining the norm behavior of non-normal matrices from the sets related to the spectrum is one of the fundamental problems of matrix theory. This article proves that the pseudospectra and condition spectra determine the norm behavior of Jordan matrices for any matrix p-norm. Further, sufficient conditions for determining the 1-norm and infinity norm behavior of bidiagonal matrices from the pseudospectra and condition spectra are also provided.

The spectral theorem implies that the spectrum of a bounded normal operator acting on a Hilbert space provides a substantial information about the operator. For example, the set of eigenvalues of a normal matrix and their respective multiplicities determine the matrix up to a unitary equivalence, while the spectral measure, (E_B(lambda )) of a normal operator B acting on a Hilbert space determines B via the integral spectral resolution,

In general, for a non-normal operator the spectrum provides a rather limited information about the operator. In this paper we show that, if we include an arbitrary bounded operator B acting on a separable Hilbert space into a quadruple which contains 3 specific operators along with B, it is possible to reconstruct B from the proper projective joint spectrum of the quadruple (and here we mean reconstruct precisely, not up to an equivalence). We call this process spectral reconstruction.