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The famous Douglas theorem contains four equivalent conditions on the solvability of the operator equation A X = B associated to Hilbert spaces, which was generalized to the Hilbert C ⁎ -module case by Fang et al. (2009) [5] . In this paper we prove that in the Hilbert C ⁎ -module case, the majorization condition and the range inclusion condition are in fact not equivalent.

In this paper we obtain some results concerning the ascent and descent of a quasi-Fredholm relation in a Hilbert space and we analyze the behaviour of a polynomial in a quasi-Fredholm relation in a Hilbert space.

The Hilbert transform is an important tool in image processing and optics. The Hilbert transform can be generalized to a fractional Hilbert transform. The generalization is driven by optics and image processing. We will generalize the fractional Hilbert transform into 2 dimensions by rotating the Hilbert transform in \({\mathbb{R}^{3}}\). The definition of the Hilbert transform as well as of the rotations will be done by quaternions.

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

In [20] and [22], the author proved that the union of two linear star-configurations in of type with and has generic Hilbert function. In this paper, we prove that the union of two linear star-configurations in of type with and has also generic Hilbert function.

We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann-Hilbert problems via Plemelj's lemma. Using this framework, we re-derive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex plane. We then demonstrate the power of this approach by generalizing to the half line. Combining two half lines, we can compute the Hilbert transform of a more general class of functions on the real line than is possible with existing methods.

We consider a completion of a space of symmetric polynomials on l1 with respect to a certain Hilbert norm. It is shown that this completion consists of analytic functions on a unit ball in l1.

It is well-known that the Ginzburg-Landau equation on $R$ has a global attractor [15] that attracts in $L^\infty_{l o c}(R)$ all the trajectories. This attractor contains bounded trajectories that are analytical functions in space. A famous theorem due to P. Collet and JP. Eckmann asserts that the $\varepsilon$-entropy per unit length in $L^\infty$ of this global attractor is finite and is smaller than the corresponding complexity for the space of functions which are analytical in a strip. This means that the global attractor is flatter than expected. We explain in this article how to establish the Collet-Eckmann Theorem in a Hilbert space framework.

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids [Formula: see text]. We also show that the Hilbert series of the positive words in [Formula: see text] with a given prefix are rational functions.

We present a new technique for computing Hilbert series of N=1 supersymmetric QCD in four dimensions with unitary and special unitary gauge groups. We show that the Hilbert series of this theory can be written in terms of determinants of Toeplitz matrices. Applying related theorems from random matrix theory, we compute a number of exact Hilbert series as well as asymptotic formulae for large numbers of colours and flavours -- many of which have not been derived before., Comment: 49 pages, 6 figures. Version 2: references added

In this paper we prove a high order Schwarz-Pick lemma for holomorphic mappings between unit balls in complex Hilbert spaces. In addition, a Schwarz-Pick estimate for high order Frechet derivatives of a holomorphic function f of a Hilbert ball into the right half-plane is obtained.

We examine condition numbers, preconditioners, and iterative methods for finite element discretizations of coercive PDEs in the context of the fundamental solvability result, the Lax-Milgram lemma. Working in this Hilbert space context is justified because finite element operators are restrictions of infinite-dimensional Hilbert space operators to finite-dimensional subspaces. Moreover, useful insight is gained as to the relationship between Hilbert space and matrix condition numbers, and translating Hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes. In this framework, the “simplest” preconditioner for an operator from a Hilbert space into its dual is the Riesz isomorphism. Simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen. Moreover, the abstraction allows us to consider not only Riesz map preconditioning for convection-diffusion equations in $H^1$ but also operators on other Hilbert spaces, such as planar elasticity in $\left(H^1\right)^2$.

Inspired by examples of small Hilbert matrices, the author proves a property of symmetric totally positive Cauchy matrices, called AT-property, and consequences for the Hilbert matrix.

We show that every simple abelian real Hilbert ternary algebra is isomorphic to the algebra of Hilbert-Schmidt operators between two real, complex or quaternionic Hilbert spaces, up to a positive multiple of the inner product.

We obtain a necessary and sufficient condition on a polynomial P(s, t) so that the (global) double Hilbert transforms along polynomial surfaces (s, t, P(s, t)) in R3 are bounded on Lp for 1 < p < ∞.

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε>0 there exists a sequence with density >1−ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.

In this paper, we introduce the notion of a norm in Hilbert algebras, and discuss some properties of Cauchy sequences.

Given a classical symmetric pair, (G,K), with g=Lie(G), we provide descriptions of the Hilbert series of the algebra of K-invariant vectors in the associated graded algebra of U(g) viewed as a K-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note that the stable Hilbert series of one symmetric pair often coincides with others. Also, for the case of the real form U(p,q) we derive a closed expression for the Hilbert series when min(p,q)→∞.

We study the tangent space at a monomial point M in the Hilbert scheme that parameterizes all ideals with the same Hilbert function as M over an exterior algebra.

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space H. We show that if W and its inverse W −1 both satisfy a matrix reverse Holder property introduced by Christ and Goldberg, then the weighted Hilbert transform H : L 2(R,H) ! L 2(R,H) and also all weighted dyadic martingale transforms T� : L 2(R,H) ! L 2(R,H) are bounded. We also show that this condition is not necessary for the boundedness of the weighted Hilbert transform.

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[ x 1 , ..., x k ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ( [Hilbert, 1890]) . In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[ x 1 , ..., x k ]/ I .

Recently, Noo and coworkers discovered an explicit inversion formula for the finite Hilbert transform, which is very important to accurate reconstruction from truncated projections. This letter presents two formulae for the finite inverse Hilbert transform using some elementary complex variable analysis. The new formulae do not contain the constant C and the singular endpoints that exist in the formula in Noo et al (2004 Phys. Med. Biol. 49 3903–23).

A finite Hilbert transformation associated with a polynomial is the analogue of a Hilbert transformation associated with an entire function, which is a generalization of the classical Hilbert transformation (cf. [4]). Hilbert's inequality, which has applications in analytic number theory (see Vaaler [11]), is closely related to the finite Hilbert transformation associated with a polynomial. In this paper, an explicit construction is given for the finite Hilbert transformation.

We present a unified representation theorem of the weighted Moore-Penrose inverse in Hilbert space. Specific expressions and computational procedures for the weighted Moore-Penrose inverse in Hilbert space can be uniformly derived.

We show that every extreme point of the unit ball of 2-homogeneous polynomials on a separable real Hilbert space is its exposed point and that the unit ball of 2-homogeneous polynomials on a non-separable real Hilbert space contains no exposed points. We also show that the unit ball of 2-homogeneous polynomials on any infinite dimensional real Hilbert space contains no strongly exposed points.

The constants A and B are called, respectively, the lower and upper bounds of the frame Φ and their ratio κ = κ(Φ) = B A is called the condition of the frame. Finally, a frame is said to be tight if κ(Φ) = 1, i.e., A = B . Here let us note the paper of Kozlov [2] in which tight frames were studied (in other terms). Obviously, any complete orthonormal system (o.n.s.) in H , as well as an arbitrary Riesz basis, is a frame (see the definition, for example, in [3, p. 17]). Frame systems are, in general, not minimal; however, some properties of orthogonal expansions remain valid for them. In particular, we use the inequality (see [1]) ∥∥∥∥ ∞ ∑

The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert spaces they induce. (C) 2001 Elsevier Science Inc. All rights reserved.

In quantum physics one uses density matrices on separable Hilbert spaces for the mathematical description of states. Often it is important to know when the image of a density matrix under a mapping is again a density matrix. We propose some new classes of mappings which map the space of density matrices on one Hilbert space H into the space of density matrices on some other Hilbert space K where K = H naturally is allowed. Such mappings may be ‘linear’ or ‘nonlinear’. In the case of finite dimensional Hilbert spaces we find all mappings of this type.

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-tl ) n for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ⊗ kRkl+i (which we denote R[l;i]) of R, in particular their Poincare series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.

In this paper, it is proved that, for any m unit vectors. x 1 …, x m in any n -dimensional real Hilbert space, there exists a unit vector x 0 such that for any y ∈ S n −1 . The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.

The main goal of this paper is to generalize the criterion for complete intersections well known in commutative algebra to the case of an arbitrary variety of PI-(super)algebras. As a consequence, formulae are obtained for computing the Hilbert series of the even and odd parts of relatively free superalgebras and free special Jordan superalgebras.

We give closed-form expressions for the average number of runs to cover an arbitrary square region in two dimensions using a Hilbert curve. The practical use of the derived formula is that it allows the estimation of the quality of the linearization obtained by using the Hilbert curve to map from a two-dimensional space to a one-dimensional space. Hilbert curves are used extensively as a basis for multi-dimensional indexing structures, and for declustering multi-dimensional data.

Let R0,n be the real Clifford algebra generated by vectors ei for i=1, 2, ⋯, n, where ei2=-1 and eiej+ejei=0 if i ≠ j, i, j=1, 2, ⋯, n. e0 is the unit element. In this article, the Hilbert Boundary Value Problem for 2-monogenic functions in the unit ball is investigated.

In this paper, we study spectral properties of generalized weighted Hilbert matrices. In particular, we establish results on the spectral norm, determinant, as well as various relations between the eigenvalues and eigenvectors of such matrices. We also study the asymptotic behaviour of the spectral norm of the classical Hilbert matrix., 17 pages, 2 figures, to appear in the Pacific Journal of Mathematics

We give a new simplified proof of two theorems of Froelich, Ingelstam, and Smiley. Our approach enables us also to generalize both of them. In the second section we prove a related theorem which requires different methods for its proof.

We show that, whenever the linear operator A is symmetric and densely defined, on a Hilbert space, then the abstract Cauchy problem \[ d d z u ( z ) = A ∗ ( u ( z ) ) ( z ∈ C ) , u ( 0 ) = x \frac {d}{{dz}}u(z) = {A^ \ast }(u(z))\quad (z \in {\mathbf {C}}),\qquad u(0) = x \] has an entire solution, for all initial data x in the image of e − A ¯ A ∗ {e^{ - \bar A{A^ \ast }}} , which is a dense set.

In this paper we characterize the Birkhoff–James orthogonality for elements of a Hilbert C∗-module in terms of states of the underlying C∗-algebra. We also show that the Birkhoff–James orthogonality in a Hilbert C∗-module over a C∗-algebra A and orthogonality with respect to the A -valued inner product coincide if and only if A is isomorphic to C . In addition, some new results concerning the case of equality in the triangle inequality for elements of a Hilbert C∗-module are obtained.