approximation error A linear operator L H H on a Hilbert space H is said to be self adjoint if for all f g H hLf gi hf Lgi It is said to be positive resp strictly positive if it is self adjoint and for all non trivial f H hLf fi resp hLf fi The next result the Spectral Theorem for compact operators see Section of for a proof will be useful in this and the next chapter Theorem Let L be a compact...

Let Ai (i = 1,2, · · · ,k) be bounded linear operators on a Hilbert space. This paper aims to show a characterization of operator order Ak Ak−1 · · · A2 A1 > 0 in terms of operator inequalities. Afterwards, an application of the characterization is given to operator equalities due to Douglas’s majorization and factorization theorem. Mathematics subject classification (2010): 47A63.

Proof of Proposition 1. Suppose an attribute value ai is not recorded in any of the data sources S1 through Sn for an entity instance. Then, from Assumptions 1 and 2 in the paper, we have P A= ai AS1 = ak AS2 = al ASn = atW i = k i = l i = t = P AS1 = ak A= ai P AS2 = al A= ai × · · ·×P ASn = at A= ai P AS1 = ak AS2 = al ASn = at P A= ai = 1−R A S1 / m− 1 1−RS2 / m− 1 × · · ·× 1−RSn / m− 1 P AS...

The response to Photofrin II-induced photosensitization on the activities of mitochondrial monoamine oxidase (MAO) and adenylate kinase (AK) were studied in order to gain further insight into site specific effects. Utilizing intact mitochondria in vitro, both MAO, located on the cytoplasmic side of the outer mitochondrial membrane, and AK, located in the intermembrane space, were inhibited by e...

where the matrix a(y) = [ak`(y)] is symmetric and positive de­ nite a(y) > ¬ I with ¬ > 0. Its entries belong to L # (Y ) (Y is the cube ]0; 2 o [ N and subscript `#’ means that the space consists of Y -periodic functions). In the sequel, we will make various further length-scale regularity assumptions on the coe ̄ cients ak` which play a central role throughout the paper. Some of them are as fo...

Let V be the C∗-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i1, . . . , im) with i1, . . . , im ∈ {1, . . . , k}, define a product of A1, . . . , Ak ∈ V by A1 ∗ · · · ∗ Ak = Ai1 . . . Aim . This includes the usual product A1 ∗ · · · ∗ Ak = A1 · · ·Ak and the Jordan triple product A ∗ ...

Abstract. If f(t) = ∑∞ k=0 akt k converges for all t ∈ IR with all coefficients ak ≥ 0, then the function f(< x,y >) is positive definite on H ×H for any inner product space H. Set K = {k : ak > 0}. We show that f(< x,y >) is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.