Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the spectrum of the multiplicative Hilbert matrix
153
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 $mathscr{H}^2$.
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.
We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.
A multiplicative Hankel operator is an operator with matrix representation $M(alpha) = {alpha(nm)}_{n,m=1}^infty$, where $alpha$ is the generating sequence of $M(alpha)$. Let $mathcal{M}$ and $mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(alpha) in mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(beta) in mathcal{M}_0$, $$|M(alpha) - N(beta)|_{mathcal{B}(ell^2(mathbb{N}))} = inf left {|M(alpha) - K |_{mathcal{B}(ell^2(mathbb{N}))} , : , K colon ell^2(mathbb{N}) to ell^2(mathbb{N}) textrm{ compact} right}.$$ Intimately connected with this result, it is then proven that the bidual of $mathcal{M}_0$ is isometrically isomorphic to $mathcal{M}$, $mathcal{M}_0^{ast ast} simeq mathcal{M}$. It follows that $mathcal{M}_0$ is an M-ideal in $mathcal{M}$. The dual space $mathcal{M}_0^ast$ is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space $H^2(mathbb{D}^d)$ of a finite polydisk.
In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory we show that $D$ has point spectrum iff the $hat A$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible then $M = U(p,q)/U(p) times U(q)$ with $p+q$ odd, and $Spec_p(D) = {0}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
