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Search: MSC category 15A60 ( Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] )

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1. CMB Online first

Moslehian, Mohammad Sal; Zamani, Ali
Exact and Approximate Operator Parallelism
Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module.

Keywords:$C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module
Categories:47A30, 46L05, 46L08, 47B47, 15A60

2. CMB 2008 (vol 51 pp. 86)

Nakazato, Hiroshi; Bebiano, Natália; Providência, Jo\ ao da
The Numerical Range of 2-Dimensional Krein Space Operators
The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.

Keywords:numerical range, generalized numerical range, indefinite inner product space
Categories:15A60, 15A63, 15A45

3. CMB 2001 (vol 44 pp. 270)

Cheung, Wai-Shun; Li, Chi-Kwong
Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$. The $c$-numerical range of an $n \times n$ matrix $A$ is defined by $$ W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\} \text{ an orthonormal basis for } \IC^n \Bigr\}, $$ and the $c$-numerical radius of $A$ is defined by $r_c (A) = \max \{|z| : z \in W_c (A)\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying $$ W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A. $$

Keywords:linear operator, numerical range (radius), block triangular matrices
Categories:15A04, 15A60, 47B49

4. CMB 2000 (vol 43 pp. 448)

Li, Chi-Kwong; Zaharia, Alexandru
Nonconvexity of the Generalized Numerical Range Associated with the Principal Character
Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider the {\it generalized matrix function} on $m\times m$ matrices $B = (b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m b_{j\sigma(j)}$ and the {\it generalized numerical range} of an $n\times n$ complex matrix $A$ associated with $d^H$ defined by $$ \wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that } X^*X = I_m\}. $$ It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$. We show that there exist normal matrices $A$ for which $\wmp(A)$ is not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove that a normal matrix $A $ with eigenvalues lying on a straight line has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some nonzero $\nu \in \IC$. These results extend those of Hu, Hurley and Tam, who studied the special case when $2 \le m \le 3 \le n$ and $H = S_m$.

Keywords:convexity, generalized numerical range, matrices

5. CMB 1998 (vol 41 pp. 105)

So, Wasin
An explicit criterion for the convexity of quaternionic numerical range
Quaternionic numerical range is not always a convex set. In this note, an explicit criterion is given for the convexity of quaternionic numerical range.

Categories:15A33, 15A60

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