In these notes we provide an introduction to compact linear operators on Banach and Hilbert spaces. These operators behave very much like familiar finite dimensional matrices, without necessarily having finite rank. For more thorough treatments, see [RS, Y]. Definition 1 Let X and Y be Banach spaces. A linear operator C : X → Y is said to be compact if for each bounded sequence {xi}i∈IN ⊂ X , t...

In a deformable template representation of observed images fundamental to mod-eling the variability of target pose is the action of the matrix Lie groups (such as SO(n)) on rigid templates. For the problem of target identiication using remote sensors , a Bayesian approach constructs a posterior distribution on the rotation group from which the conditional expectations can be generated. Due to t...

in the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. in particular, it isshown that the cauchy-schwarz inequality holds. moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy hilbert space has a complementary subspace.finally, the notions of fuzzy bo...

This paper aims to study the α-volume of K, an arbitrary subset of the set of N ×N density matrices. The α-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the α-volume of K in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the α-volume are also established. We employ our...

Abstract. We first show that the canonical solution operator to ∂ restricted to (0, 1)forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in C the canonical solution operator to ∂ restricted to (0, 1)-forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel w...

Conjecture 0.1. Do the following conditions: vn → 0 and vn+q − vn ∈ l (q ∈ Z– fixed) guarantee that σac(J) = [−2, 2]? The symbol σac(J) conventionally denotes the absolutely continuous (a.c.) spectrum of self-adjoint operator J . In this paper, we give an affirmative answer to this question. The manuscript consists of two sections. The first one is mostly algebraic, it contains the determinanta...

We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products.

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the rever...

In this paper, a novel semi-supervised dictionary learning and sparse representation (SS-DLSR) is proposed. The proposed method benefits from the supervisory information by learning the dictionary in a space where the dependency between the data and class labels is maximized. This maximization is performed using Hilbert-Schmidt independence criterion (HSIC). On the other hand, the global distri...

In recent work by (Song et al., 2007), it has been proposed to perform clustering by maximizing a HilbertSchmidt independence criterion with respect to a predefined cluster structure Y , by solving for the partition matrix, Π. We extend this approach here to the case where the cluster structure Y is not fixed, but is a quantity to be optimized; and we use an independence criterion which has bee...