Karl-Mikael Perfekt

The multiplicative Hilbert matrix

Abstract It is observed that the infinite matrix with entries ( m n log ⁡ ( m n ) ) − 1 for m , n ≥ 2 appears as the matrix of the integral operator H f ( s ) : = ∫ 1 / 2 + ∞ f ( w ) ( ζ ( w + s ) − 1 ) d w with respect to the basis ( n − s ) n ≥ 2 ; here ζ ( s ) is the Riemann zeta function and H is defined on the Hilbert space H 0 2 of Dirichlet series vanishing at +∞ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helsons terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus T ∞ . By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals π and that it has a purely continuous spectrum which is the interval [ 0 , π ] ; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix ( m 1 / p n ( p − 1 ) / p log ⁡ ( m n ) ) − 1 has norm π / sin ⁡ ( π / p ) when acting on l p for 1 p ∞ . However, the multiplicative Hilbert matrix fails to define a bounded operator on H 0 p for p ≠ 2 , where H 0 p are H p spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol ∑ n ≥ 2 ( log ⁡ n ) − 1 n − s − 1 / 2 of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus T ∞ .

The Essential Spectrum of the Neumann–Poincaré Operator on a Domain with Corners

Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors–Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors–Beurling transform and the Neumann–Poincaré operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann–Poincaré operator on a planar domain with corners, with respect to the energy norm of the associated harmonic field.

Duality and distance formulas in spaces defined by means of oscillation

For the classical space of functions with bounded mean oscillation, it is well known that

Operator-Lipschitz estimates for the singular value functional calculus

\operatorname{VMO}^{**} = \operatorname{BMO}

On Helson matrices: moment problems, non‐negativity, boundedness, and finite rank

and there are many characterizations of the distance from a function f in

Trace ideal criteria for embeddings and composition operators on model spaces

\operatorname{BMO}

Volterra operators on Hardy spaces of Dirichlet series

to

Failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes

\operatorname{VMO}

Endpoint compactness of singular integrals and perturbations of the Cauchy integral

. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as QK-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular

Weak Product Spaces of Dirichlet Series

\operatorname{BMO}