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تسجيل مستخدم جديدOn some 4-by-4 matrices with bi-elliptical numerical ranges
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A complete description of 4-by-4 matrices $begin{bmatrix}alpha I & C D & beta Iend{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity. In particular cases when in addition $alpha-beta$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.
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The 4-by-4 nilpotent matrices the numerical ranges of which have non-parallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin
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This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.
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e unitarily irreducible. The Gau--Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalar products $langle Ax_j,x_jrangle$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4times4$ matrices according to the values of their Gau--Wu numbers.
The center of mass of an operator $A$ (denoted St($A$), and called in this paper as the {em Stampfli point} of A) was introduced by Stampfli in his Pacific J. Math (1970) paper as the unique $lambdainmathbb C$ delivering the minimum value of the norm
of $A-lambda I$. We derive some results concerning the location of St($A$) for several classes of operators, including 2-by-2 block operator matrices with scalar diagonal blocks and 3-by-3 matrices with repeated eigenvalues. We also show that for almost normal $A$ its Stampfli point lies in the convex hull of the spectrum, which is not the case in general. Some relations between the property St($A$)=0 and Roberts orthogonality of $A$ to the identity operator are established.
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