Superspecial abelian varieties and the Eichler Basis Problem for Hilbert modular forms

Let p be an unramified prime in a totally real field L such that h⁺(L)=1. Our main result shows that Hilbert modular newforms of parallel weight two for Γ₀(p) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.     

Factorization of J-expansive meromorphic operator-valued functions

The factorization theorems are a generalization for J-biexpansive meromorphic operator-valued functions on an infinite-dimensional Hilbert space of the theorems on decomposition of J-expansive matrix functions on a finite-dimensional Hilbert space due to A. V. Efimov and V. P. Potapov [Uspekhi Mat. Nauk28 (1973), 65–130; Trudy Moskov. Mat. Ob??.4 (1955), 125–236]. They also generalize theorems on factorization of J-expansive meromorphic operator functions due to Ju. P. Ginzburg [Izv. Vys?. U?ebn. Zaved. Matematika32 (1963), 45–53]. Within the framework of generalized network theory, the results can be applied to the J-biexpansive real operators that characterize a Hilbert port. Application of the extraction procedure to a given real operator leads to its splitting into a product of real factors, corresponding to Hilbert ports of a simpler structure. This can be interpreted as an extension of the classical method of synthesis of passive n-ports by factor decomposition.     

Extension of Gaussian cylinder set measures

Let E be a one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K, with E having dense range. One considers Gaussian cylinder set measures on H defined by weak covariance operators. Such cylinder set measures may be used to induce, through E, Gaussian cylinder set measures on K. The result of this paper extends a result of Sato: it characterizes the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K. Such a result is of interest in the robustness study of signal detection.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

An analog of η(z) in the Hilbert modular case

Certain Hilbert modular forms of weight $\text{1}\text{2}$, analogous to η(z), with respect to the full Hilbert modular group SL₂(°) are constructed, where o is the maximal order of a totally real algebraic number field.     

Linear polarization constants of Hilbert spaces

This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant cn(X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that cn(X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then cn(H)=nn/2. The same value of cn(H) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order dn/2, for n large enough, and also an integral form of the asymptotic quantity c(H), that is the (linear) polarization constant of the Hilbert space H, where dimH=d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of cn(H) and c(H). In particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999) 315-322]. Also, we prove the conjectured cn(H)=nn/2 for real Hilbert spaces of dimension n?5. A few further estimates have been also derived.     

Overconvergent modular sheaves and modular forms for GL 2/F

Given a totally real field F and a prime integer p which is unramified in F, we construct p-adic families of overconvergent Hilbert modular forms (of non-necessarily parallel weight) as sections of, so called, overconvergent Hilbert modular sheaves. We prove that the classical Hilbert modular forms of integral weights are overconvergent in our sense. We compare our notion with Katz’s definition of p-adic Hilbert modular forms. For F = ?, we prove that our notion of (families of) overconvergent elliptic modular forms coincides with those of R. Coleman and V. Pilloni.     

Explicit determination of images of Galois representations attached to Hilbert modular forms

In a previous paper the second author proved that the image of the Galois representation modulo λ attached to a Hilbert modular newform is “large” for all but finitely many primes λ, if the newform is not a theta series. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or non-parallel weight and of different levels.     

Description of Real AW*-Factors of Type I

In the paper, real AW*-algebras are considered, i.e., real C*-algebras which are Baer *-rings. It is proved that every real AW*-factor of type I (i.e., having a minimal projection) is isometrically *-isomorphic to the algebra B(H) of all bounded linear operators on a real or quaternionic Hilbert space H and, in particular, is a real W*-factor. In the case of complex AW*-algebras, a similar result was proved by Kaplansky.     

Asymptotics of the n-widths of a Wiener spiral in complex Hilbert space

We describe isometric embeddings of the Wiener spiral in complex Hilbert space and obtain the asymptotics of the Kolmogorov n-widths of specific embeddings. We note the difference with the asymptotics of the n-widths of the Wiener spiral in real Hilbert space.     

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C＊-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.     

Isometric reflection vectors in Banach spaces

The aim of this paper is to study the set of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every real Banach space can be decomposed as an Ir-sum of a Hilbert space and a Banach space with some points which are not isometric reflection vectors. Finally, we give a new proof of the Becerra-Rodríguez result: a real Banach space X is a Hilbert space if and only if is not rare.     

Central values of L-functions over CM fields

We prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are the L-functions attached to Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. This formula generalizes our former result on L-functions twisted by finite CM characters.     

Two examples on the convergence of certain rank-2 minimization methods for quadratic functionals in Hilbert space

Two simple examples are given showing that the usual rank-2 algorithms for minimizing functionals $\text{?:H→}\text{R}$ on a real Hilbert space H may converge only linearly and in particular bad cases only sublinearly, even for quadratic ?.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

Elementary localization theorems for nonlinear eigenproblems

The minimax formula for linear eigenvalues of a linear operator is used to estimate the parameter values (λ) for which the self-adjoint operator L(λ) on Hilbert space to itself fails to have a bounded inverse. Such λ compose the “nonlinear spectrum” of L. The parameter spaces include regions in real or complex n-space. The localization theorems are used to demonstrate certain necessary conditions for stability of linear integro-partial-differential delay equations.     

Ranges of nonlinear asymptotically linear operators

This paper deals with the solvability of the equation A(u) ? S(u) = f, where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H, the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.     

PP* estimates for the truncated Carleson operator

We consider two Hilbert transforms, with real phases a and b, smoothly truncated at ${{2^{-k_0}}}$ where k ₀ is a non-negative integer. We decode the operator resulting from composing one of them with the adjoint of the other one. Then the case of a similarly truncated Carleson operator is dealt with. The case of Hilbert transforms, as well as the Carleson operator, truncated away from the origin is also considered. An application is mentioned.     

On the stability of the Ritz procedure for nonlinear problems

In this paper we establish the stability of the Ritz procedure for the nonlinear equationAu+F(u)?f=0 whereA is a linear positive definite operator in a real Hilbert spaceH andF is a monotone nonlinear operator.     

Generalizations of Bohr inequality for Hilbert space operators

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A,B∈B(H) and p,q>1 real numbers such that 1/p+1/q=1,

²|A+B|?p²|A|+q²|B|