Integral Equations and Operator Theory最新文献

Abstract

In the context of operator valued W (^*) -free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as vx where x is self-adjoint and v is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that v and x are (*) -free from each other. In particular, we prove, when (B={textbf{C}}^2) , that if a B-valued circular element has a free bipolar decomposition with v unitary, then it has one where v normalizes B.

In this paper, we study the regularity of (mathbb {R})-differentiable functions on open connected subsets of the scaled hypercomplex numbers (left{ mathbb {H}_{t}right} _{tin mathbb {R}}) by studying the kernels of suitable differential operators (left{ nabla _{t}right} _{tin mathbb {R}}), up to scales in the real field (mathbb {R}).

Abstract

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems.

Given a probability measure (mu ) on the unit circle ({mathbb {T}}), consider the reproducing kernel (k_{mu ,n}(z_1, z_2)) in the space of polynomials of degree at most (n-1) with the (L^2(mu ))–inner product. Let (u, v in {mathbb {C}}). It is known that under mild assumptions on (mu ) near (zeta in mathbb {T}), the ratio (k_{mu ,n}(zeta e^{u/n}, zeta e^{v/n})/k_{mu ,n}(zeta , zeta )) converges to a universal limit S(u, v) as (n rightarrow infty ). We give an estimate for the rate of this convergence for measures (mu ) with finite logarithmic integral.

The purpose of this paper is to revisit the proof of the Gearhardt–Prüss–Huang–Greiner theorem for a semigroup S(t), following the general idea of the proofs that we have seen in the literature and to get an explicit estimate on the operator norm of S(t) in terms of bounds on the resolvent of the generator. In Helffer and Sjöstrand (From resolvent bounds to semigroup bounds. ArXiv:1001.4171v1, math. FA, 2010) by the first two authors, this was done and some applications in semiclassical analysis were given. Some of these results have been subsequently published in three books written by the two first authors Helffer (Spectral theory and its applications. Cambridge University Press, Cambridge, 2013) and Sjöstrand (Lecture notes : Spectral properties of non-self-adjoint operators. Journées équations aux dérivées partielles (2009), article no. 1), (Non self-adjoint differential operators, spectral asymptotics and random perturbations. Pseudo-differential Operators and Applications, Birkhäuser (2018)). A second work Helffer and Sjöstrand (Integral Equ Oper Theory 93(3), 2021) presents new improvements partially motivated by a paper of Wei (Sci China Math 64:507–518, 2021). In this third paper, we continue the discussion on whether the aforementioned results are optimal, and whether one can improve these results through iteration. Numerical computations will illustrate some of the abstract results.

We study the real and imaginary parts of the powers of the Volterra operator on (L^2[0,1]), specifically their eigenvalues, their norms and their numerical ranges.

Let (B subseteq A) be an inclusion of (C^*)-algebras. We study the relationship between the regular ideals of B and regular ideals of A. We show that if (B subseteq A) is a regular (C^*)-inclusion and there is a faithful invariant conditional expectation from A onto B, then there is an isomorphism between the lattice of regular ideals of A and invariant regular ideals of B. We study properties of inclusions preserved under quotients by regular ideals. This includes showing that if (D subseteq A) is a Cartan inclusion and J is a regular ideal in A, then (D/(Jcap D)) is a Cartan subalgebra of A/J. We provide a description of regular ideals in the reduced crossed product of a C(^*)-algebra by a discrete group.

We consider the indefinite Sturm–Liouville differential expression

where ({mathfrak {a}}) is defined on a finite or infinite open interval I with (0in I) and the coefficients r and w are locally summable and such that r(x) and (({text {sgn}},x) w(x)) are positive a.e. on I. With the differential expression ({mathfrak {a}}) we associate a nonnegative self-adjoint operator A in the Krein space (L^2_w(I)) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space (L^2_{|w|}(I)). These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.

We characterize the integration operators (V_g) with symbol g for which (V_g) acts as an absolutely summing operator on weighted Bloch spaces (mathcal {B}^{beta }) and on weighted Bergman spaces (mathscr {A}^p_alpha ). We show that (V_g) is r-summing on (mathscr {A}^p_alpha ), (1 le p <infty ), if and only if g belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators (V_g) on Bloch spaces and on Bergman spaces.

In this paper, we first define the phases of a sectorial operator based on the numerical range. We are interested in the estimation of the spectrum phases of AB based on the phases of two sectorial operators A and B. Motivated by the classical small gain theorem, we formulate an operator small phase theorem with necessity for the invertibility of (I+AB), which plays a crucial role in feedback stability analysis. Afterwards, we consider the special class of sectorial operators of the form (P+K), where P is strictly positive and K is compact. More properties of the phases for those operators are studied, including those of compressions, Schur complements, operator means and products. Finally, for the special class of sectorial operators, we further establish a majorization relation between the phases of the spectrum of AB and the phases of two operators A and B.