Superpositions for nonlinear operators II. Generalized affine operators

From a previously given type of operators having strong superpositions, the C_f1,f2 classes, a generalization of affine transformations is obtained. The resulting operators have invariant quasilinear means. In addition, the operators have strong superpositions which are abelian semigroup operations with an idempotent property. It is natural, in this case, to define scalar operations on pairs of scalars and pairs of vectors on the domain and range spaces. Properties of this algebraic structure and its similarity to the superposition rules for color sensations are shown.     

Models for n-tuples of noncommuting operators

This paper proves versions of the Rota model theorem, the de Branges-Rovnyak model theorem, and the coisometric extension theorem for n-tuples of not necessarily commuting operators. This generalizes the work of A. E. Frazho (J. Funct. Anal.48 (1982), 1–11) for pairs of operators. The methods involve applying the single operator results to matrices of operators.     

Decomposition formulas for some triple hypergeometric functions

With the help of some techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Srivastava's hypergeometric functions HA, HB and HC in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 15 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly.     

Extreme positive operators on minimal and almost minimal cones

Fiedler and Pták called a cone minimal if it is n-dimensional and has n+1 extremal rays. We call a cone almost minimal if it is n-dimensional and has n+2 extremal rays. Duality properties stemming from the use of Gale pairs lead to a general technique for identifying the extreme cone-preserving (positive) operators between polyhedral cones. This technique is most effective for cones with dimension not much smaller than the number of their extreme rays. In particular, the Fiedler-Pták characterization of extreme positive operators between minimal cones is extended to the following cases: (i) operators from a minimal cone to an arbitrary polyhedral cone, (ii) operators from an almost minimal cone to a minimal cone.     

Which linear-fractional composition operators are essentially normal?

We characterize the essentially normal composition operators induced on the Hardy space H² by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H² that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.     

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325–326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators.     

A Generalization of the Heine�CStieltjes Theorem

The Heine?CStieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.     

Quadratic forms and Klauder's phenomenon: A remark on very singular perturbations

We give a general analysis of a class of pairs of positive self-adjoint operators A and B for which A + λB has a limit (in strong resolvent sense) as λ ↓ 0 which is an operator A_p ≠ A!     

Computations of cuspidal cohomology of congruence subgroups of SL(3, Z)

Algorithms are presented which find a basis of the vector space of cuspidal cohomology of certain congruence subgroups of SL(3, $\text{Z}$) and which determine the action of the Hecke operators on this space. These algorithms were implemented on a computer. Four pairs of cuspidal classes were found with prime level less than 100. Tables are given of the eigenvalues of the first few Hecke operators on these classes.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

On linear preservers of normal maps

We study group induced cone (GIC) orderings generating normal maps. Examples of normal maps cover, among others, the eigenvalue map on the space of n × n Hermitian matrices as well as the singular value map on n × n complex matrices. In this paper, given two linear spaces equipped with GIC orderings induced by groups of orthogonal operators, we investigate linear operators preserving normal maps of the orderings. A characterization of the preservers is obtained in terms of the groups. The result is applied to show that the normal structure of the spaces is preserved under the action of the operators. In addition, examples are given.     

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of \(SL(2,\mathbb {R})\). We address a general problem to find explicit formulæ  for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9) and based on the algebraic Fourier transform for generalized Verma modules.The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulæ  of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.     

A simple averaging process for approximating the solutions of certain optimal control problems

It is shown that the extremal solutions of fixed duration Mayer control problems with implicit terminal constraints can be interpreted as fixed points of certain function-valued operators F constructed by solving pairs of initial value problems in tandem. A class of simple recursive averaging processes is proposed for approximating the fixed points of F. Results from the theory of monotone Hilbert space operators are used to establish the convergence of the averaging processes for a general linear-quadratic curve follower problem with unbounded control inputs, and for a simple second order bounded control input problem.     

Schur product techniques for commuting multivariable weighted shifts

In this paper we study the hyponormality and subnormality of 2-variable weighted shifts using the Schur product techniques in matrices. As applications, we generalize the result in [R. Curto, J. Yoon, Jointly hyponormal pairs of subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc. 358 (2006) 5135-5159, Theorem 5.2] and give a non-trivial, large class satisfying the Curto-Muhly-Xia conjecture [R. Curto, P. Muhly, J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35 (1988) 1-22] for 2-variable weighted shifts. Further, we give a complete characterization of hyponormality and subnormality in the class of flat, contractive, 2-variable weighted shifts T≡(T₁,T₂) with the condition that the norm of the 0th horizontal 1-variable weighted shift of T is a given constant.     

Holomorphic extensions of Laplacians and their determinants

The Teichmüller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det^′(Δ) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {Δμ,ν} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant of this family {Δμ,ν} and show how this family realizes the holomorphic extension of det^′(Δ) as its determinant.     

Reflexive *-derivations and lattices of invariant subspaces of operator algebras associated with them

The paper studies unbounded reflexive *-derivations δ of C*-algebras of bounded operators on Hilbert spaces H whose domains D(δ) are weekly dense in B(H and contain compact operators. It describes a one-to-one correspondence between these derivations and pairs S,L, where S are symmetric densely operators on H and L are J-orthogonal π-reflexive lattices of subspaces in the deficiency spaces of S. The domains D(δ) of these *-derivations are associated with some non-selfadjoint reflexive algebras A_δ of bounded operators on H⊕H. The paper analyzes the structure of the lattices of invariant subspaces of A_δ and of the normalizers of A_δ-the largest Lie subalgebras of B(H⊕H) such that A_δ are their Lie ideals.     

Spin Leonard pairs

Let ${\mathbb K}$ denote a field, and let V denote a vector space over ${\mathbb K}$ of finite positive dimension. A pair A, A* of linear operators on V is said to be a Leonard pair on V whenever for each B∈{A, A*}, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A* on V is said to be a spin Leonard pair whenever there exist invertible linear operators U, U* on V such that UA = A U, U*A* = A*U*, and UA* U ^?1 = U*^?1 AU*. In this case, we refer to U, U* as a Boltzmann pair for A, A*. We characterize the spin Leonard pairs. This characterization involves explicit formulas for the entries of the matrices that represent A and A* with respect to a particular basis. The formulas are expressed in terms of four algebraically independent parameters. We describe all Boltzmann pairs for a spin Leonard pair in terms of these parameters. We then describe all spin Leonard pairs associated with a given Boltzmann pair. We also describe the relationship between spin Leonard pairs and modular Leonard triples. We note a modular group action on each isomorphism class of spin Leonard pairs.     

On normal τ-measurable operators affiliated with semifinite von Neumann algebras

Let τ be a faithful normal semifinite trace on the von Neumann algebra M, 1 ≥ q > 0. The following generalizations of problems 163 and 139 from the book [1] to τ-measurable operators are obtained; it is established that: 1) each τ-compact q-hyponormal operator is normal; 2) if a τ-measurable operator A is normal and, for some natural number n, the operator A ⁿ is τ-compact, then the operator A is also τ-compact. It is proved that if a τ-measurable operator A is hyponormal and the operator A ² is τ-compact, then the operator A is also τ-compact. A new property of a nonincreasing rearrangement of the product of hyponormal and cohyponormal τ-measurable operators is established. For normal τ-measurable operators A and B, it is shown that the nonincreasing rearrangements of the operators AB and BA coincide. Applications of the results obtained to F-normed symmetric spaces on (M, τ) are considered.     

Perturbation theory for commutative m-tuples of self-adjoint operators

Generalizing the Weyl-von Neumann theorem for normal operators, we show that a commutative m-tuple of self-adjoint operators in a separable Hilbert space may be changed into a diagonal one by adding compact perturbations of class c_p, for p>m. On the other hand it is shown that the absolutely continuous part, defined appropriately, of a commutative m-tuple of self-adjoint operators is stable under perturbations of class c_p, if p < m, m ? 3, or if p = 1, m = 2 (the latter case m = 2 corresponding to the case of one normal operator). For the proof of these Kato-Rosenblum-type theorems a wave operator method for m-tuples is introduced.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.