We first introduce some related definitions of the bounded linear operator L in the reproducing kernel space W(2)(m)(D). Then we show spectral analysis of L and derive several property theorems.

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

In this paper, we consider the optimal disturbance rejection problem for (possibly infinite dimensional) linear time-varying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. After reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, whic...

Let $\mathcal H $ be a complex, separable Hilbert space, and B(\mathcal H) denote the set of all bounded linear operators on $. Given an orthogonal projection $P \in \mathcal operator $D B(\mathc

An analytic self-map φ : D → D of the open unit disk D in the complex plane induces the composition operator Cφ on H(D), the space of holomorphic functions on D, defined by Cφ(f) = f ◦φ. A basic goal in the study of composition operators is to relate function theoretic properties of φ to operator theoretic properties of Cφ. Here we review some results that characterize when φ induces a bounded ...

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...

Let Xo and X, be two Banach spaces which are continuously embedded in a common Hausdort t topological vector space. An admissible operator for the pair (Xo, X,) is a linear operator whose domain contains the union of the two spaces and whose restrictions to Xi is a bounded operator on Xi ( i = 0 , 1). A space X is called an interpolation space for the pair (Xo, X,) if each admissible operator T...

Let L be a bounded linear Fredholm mapping of index zero, mapping a Banach space X into a Banach space 7. Then necessary and sufficient conditions for the solvability of the operator equation Lu = ƒ for ƒ e Y are well known. However the same satisfactory state of affairs does not hold for semilinear operator equations in which a compact nonlinear operator Nu is added to the right hand side of L...

We prove that if q is in (1,∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in R with the property that sup{‖Ta‖Y : a is a (1, q)-atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H(R) to Y . We show that the same is true if we replace (1, q)-atoms by continuous (1,∞)-atoms. This is k...

We investigate existence of weak solutions and their smoothness properties for a type of generalized Stokes problem. The generalization we consider here consists in two points: A Laplacean is replaced by a general second order linear elliptic operator in divergence form and “pressure” gradient ∇p is replaced by a linear operator of the first order. Introduction Results contained in this contrib...