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Register a new userThe spectrum of some Hardy kernel matrices
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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 $mathscr{H}^2$.
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We obtain Sobolev inequalities for the Schrodinger operator -Delta-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel, improving upon earlier results.
We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(sqrt{mn}log(mn))^{-1}$ for $m,ngeq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,pi]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.
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.
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 study generalized polar decompositions of densely defined, closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.
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