Integral Equations and Operator Theory最新文献

Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with concepts from tropical geometry, proving that tropical positive semidefinite kernels are also endowed with equivalent viewpoints, stemming from Fenchel–Moreau conjugations. This tropical analogue of Aronszajn’s theorem shows that these kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. However, two relevant notions of tropical reproducing kernels must be distinguished, based either on linear or sesquilinear interpretations. The sesquilinear interpretation is the most expressive one, since reproducing spaces then encompass classical max-plus spaces, such as those of (semi)convex functions. In contrast, in the linear interpretation, the reproducing kernels are characterized by a restrictive condition, von Neumann regularity. Finally, we provide a tropical analogue of the “representer theorems”, showing that a class of infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We illustrate this theorem by an application to optimal control, in which tropical kernels allow one to represent the value function.

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of “selfadjoint” interface conditions, subject to an assumption which is generically satisfied, is considered. We determine the spectral multiplicity function on the singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.

We offer a new perspective and some advances on the 1971 Pearcy-Topping problem: is every compact operator a commutator of compact operators? Our goal is to analyze and generalize the 1970’s work in this area of Joel Anderson. We reduce the general problem to a simpler sequence of finite matrix equations with norm constraints, while at the same time developing strategies for counterexamples. Our approach is to ask which compact operators are commutators AB − BA of compact operators A,B; and to analyze the implications of Joel Anderson’s contributions to this problem. By extending the techniques of Anderson, we obtain new classes of operators that are commutators of compact operators beyond those obtained by the second and the fourth author. We also found obstructions to extending Anderson’s techniques to obtain any positive compact operator as a commutator of compact operators. Some of these constraints involve general block-tridiagonal matrix forms for operators and some involve B(H)-ideal constraints. Finally, we provide some necessary conditions for the Pearcy-Topping problem involving singular numbers and B(H)-ideal constraints.

A unital (C^*)-algebra is called N-subhomogeneous if its irreducible representations are finite dimensional with dimension at most N. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering (text {UCP }(mathcal {S})) which is the matrix state space associated with an operator system (mathcal {S}) and identifying the boundary representations as absolute matrix extreme points. We show that two N-subhomogeneous operator systems are completely order equivalent if and only if they are N-order equivalent. Moreover, we show that a unital N-positive map into a finite dimensional N-subhomogeneous operator system is completely positive. We apply these tools to classify pairs of q-commuting unitaries up to (*)-isomorphism. Similar results are obtained for operator systems related to higher dimensional non-commutative tori.

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and (C^*)-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in (L_p(mathbb {R}^d)) for (pin [1,infty )) and thick control sets.

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra (mathcal {N}mathcal {T}(X)) of a product system (A, X) over (mathbb {N}^n). We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in (mathcal {Nhspace{-1.111pt}T}(X)) to A. The main result is a classification of gauge-invariant ideals in (mathcal {Nhspace{-1.111pt}T}(X)) for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.

We show how to find the shape of an equilateral tree using the spectra of the Neumann and the Dirichlet problems generated by the Sturm–Liouville equation. In case of snowflake trees the spectra of the Neumann and Dirichlet problems uniquely determine the shape of the tree.