Let A=(A1, . . ., Am) be an m-tuple of n × n Hermitian matrices. For $1 \le k \le n$, the $k${\rm th} joint numerical range of A is defined by $$W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.$$ We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so that Wk(A) is convex. It is shown that m can reach the upper bound 2k(n-k)+1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

background

/pdf/convexity-of-the-joint-numerical-range-33sk0ug53w.pdf

Some classes of commuting n-tuples of operators

Convexity of the Joint Numerical Range

/pdf/spectral-properties-of-p-hyponormal-operators-5go5hyq9q6.pdf

Spectral properties of p -hyponormal operators

For every m-tuple of operators acting on a Hilbert space, it is shown that there exists a common dilation of these operators to mcommuting normal operators on some larger Hilbert space. We then introduce a norm on the m-fold cartesian product of ℬ(ℋ) that is defined to be, for a given w-tuple, the infimum of the joint spectral radii of all joint normal dilations of the m operators. This norm has several good features, one of which is that it is invariant under the passage to adjoints.

A Dilation and Norm in Several Variable Operator Theory

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

/pdf/joint-spectra-and-joint-numerical-ranges-for-pairwise-3lcnoogq9q.pdf

Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

LetHn be the real linear space of n n complex Hermitian matrices. The unitary (similarity) orbitU(C) of C 2 Hn is the collection of all matrices unitarily similar to C. We characterize those C 2 Hn such that every matrix in the convex hull ofU(C) can be written as the average of two matrices inU(C). The result is used to study spectral properties of submatrices of matrices inU(C), the convexity of images ofU(C) under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

http://cms.math.ca/cjm/abstract/pdf/149526.pdf

Some Convexity Features Associated with Unitary Orbits

/pdf/some-classes-of-commuting-n-tuples-of-operators-1j63w9xbnq.pdf

Some classes of commuting n-tuples of operators

Citations

Convexity of the Joint Numerical Range

Spectral properties of p -hyponormal operators

A Dilation and Norm in Several Variable Operator Theory

Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

Some Convexity Features Associated with Unitary Orbits

Related Papers (5)

A joint spectrum for several commuting operators

On a question of N. Salinas

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ON JOINT HYPONORMALITY OF COMMUTING n-TUPLES OF OPERATORS