Makoto Takaguchi

Bio: Makoto Takaguchi is an academic researcher. The author has contributed to research in topics: Tuple & Joint (geology). The author has an hindex of 3, co-authored 3 publications receiving 46 citations.

Convexity of the Joint Numerical Range

Abstract: 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.

Joint normality of operators in semi-Hilbertian spaces

Abstract: In this paper, we introduce the concept of normality of a d-tuple of bounded linear operators acting on a complex Hilbert space H when an additional semi-inner product induced by a positive operato...

The joint essential numerical range of operators: convexity and related results

Abstract: LetW (A) andWe(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A1, . . . , Am) acting on an infinite dimensional Hilbert space, respectively. In this paper, it is shown that We(A) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ {1, . . . ,m}, We(A) can be obtained as the intersection of all sets of the form cl(W (A1, . . . , Ai+1, Ai + F,Ai+1, . . . , Am)), where F = F ∗ has finite rank Moreover, it is shown that the closure cl(W (A)) of W (A) is always star-shaped with the elements in We(A) as star centers. Although cl(W (A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d / ∈ cl(W (A)), there is a linear functional f such that f(d) > sup{f(a) : a ∈ cl(W (A))}, where A is obtained from A by perturbing one of the components Ai by a finite rank self-adjoint operator. Other results on W (A) and We(A) extending those on a single operator are obtained. AMS Subject Classification 47A12, 47A13, 47A55.