Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra a metatime with measure and defining composition given by on domain σ-algebra, we identify metatimes bounded linear operators spaces square integrable functions. We also analyse possibility define from operator Hilbert which show is possible fo...

We study fundamental solutions to second order parabolic systems of divergence type with time independent coefficients, and give another proof of a result by Auscher, McIntosh and Tchamitchian on the Gaussian bounds for the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients.

We prove the Kato conjecture for elliptic operators on Rn. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = −div (A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2.

The Brudnyi-Krugljak theorem on the if-divisibility of Gagliardo couples is derived by elementary means from earlier results of LorentzShimogaki on equimeasurable rearrangements of measurable functions. A slightly stronger form of Calderón's theorem describing the Hardy-LittlewoodPólya relation in terms of substochastic operators (which itself generalizes the classical Hardy-Littlewood-Pólya re...

We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions: (i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the W -type (in particular, acting on the time-variables in the HurwitzKontsevich ...

We study Lp-theory of second order elliptic divergence type operators with measurable coefficients. To this end, we introduce a new method of constructing positive C0-semigroups on Lp associated with sesquilinear (not necessarily sectorial) forms in L2. A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C0-semigroup on Lp is established.

We investigate the Lp-spectrum of linear operators defined consistently on Lp(Ω) for p0 ≤ p ≤ p1, where (Ω, μ) is an arbitrary σ-finite measure space and 1 ≤ p0 < p1 ≤ ∞. We prove p-independence of the Lp-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω, μ); the balls with respect to this semi-metric are required to satisfy a...

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theore...

We prove a “local” Tb Theorem for square functions, in which we assume only Lq control of the pseudo-accretive system, with q > 1. We then give an application to variable coefficient layer potentials for divergence form elliptic operators with bounded measurable non-symmetric coefficients.

In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, wh...