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Tridiagonal matrices with constant main diagonal and reciprocal pairs of off-diagonal entries are considered. Conditions for such matrices with sizes up to 6-by-6 to have elliptical numerical ranges are obtained.
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The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.
By definition, reciprocal matrices are tridiagonal $n$-by-$n$ matrices $A$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,ldots,n-1$. For $nleq 6$, we establish criteria under which the numerical range generating curves (also called Kippenhahn curves) of such matrices consist of elliptical components only. As a corollary, we also provide a complete description of higher-rank numerical ranges when the criteria are met.
For $alpha > 0$ we consider the operator $K_alpha colon ell^2 to ell^2$ corresponding to the matrix [left(frac{(nm)^{-frac{1}{2}+alpha}}{[max(n,m)]^{2alpha}}right)_{n,m=1}^infty.] By interpreting $K_alpha$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2/alpha]$ (multiplicity one), and that there is no singular continuous spectrum. There is a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series $mathscr{H}^2$.
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.
We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_beta$ process, which describes the largest eigenvalues in the $beta$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ram{i}rez, Rider, and Vir{a}g. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.
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