Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

Abstract

We present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.

On interplay between operators, bases, and matrices

“…See also an open question on [56, p. 45] asking for a version of Theorem 2.5. Generalizations of (1.3) in the framework of diagonals for operator tuples, and also to the context of operator-valued diagonals, can be found in [46, Theorems 1.2 and 1.3], see also [47].…”

“…For the theory of essential numerical ranges of operator tuples, one may consult e.g. [37], [45], [43] and [47].…”

“…A nice survey of recent developments in the theory of operator diagonals can be found in [38]. See also [46] and a discussion of some recent applications of numerical ranges in [47], including the references therein.…”

On the Joint A-Numerical Radius of Operators and Related Inequalities

“…Additionally, intrinsic problems, such as the convexity of various types of generalized numerical ranges, the realizability of certain sets (such as the numerical ranges of an operator), the completability of partial matrices, and the classification of linear preservers are of interest. For more information on some of these applications, interested readers may refer to the following references, such as [11,12], and the references within. The applications mentioned above have motivated us to explore the connection between the Ajoint numerical radius of operators and other areas of applied mathematics.…”

The joint numerical range of commuting matrices

“…, A m } ⊆ M n , and has applications in many pure and applied areas. We refer the readers to the excellent survey [14] and the paper [15] on this subject.…”