Semivariation and operator semivariation of Hilbert space valued measures

Operator-valued bases on Hilbert spaces

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...

On Maximal Regularity and Semivariation of α-Times Resolvent Families*

Let and A be the generator of an -times resolvent family 1 α < < 2 α ( ) { } 0 t S t α ≥ on a Banach space X. It is shown that the fractional Cauchy problem , ( ) ( ) ( ) t u t Au t f t α = + D ( ] 0, t r ∈ ; has maximal regularity on ( ) ( 0 , u u′ ) ( ) 0 D A ∈ [ ] ( 0, ; C r ) X if and only if is of bounded semivariation on ( ) ⋅ Sα [ ] 0, r .

Remarks on the Semivariation of Vector Measures with Respect to Banach Spaces

Let L(ν)⊗̂γqY = Lq(ν, Y ) and X⊗̂∆pL(μ) = Lp(μ,X). It is shown that any Lp(μ)-valued measure has finite L2(ν)-semivariation with respect to the tensor norm L(ν)⊗̂∆pL(μ) for 1 ≤ p < ∞ and finite Lq(ν)semivariation with respect to the tensor norm L(ν)⊗̂γqL(μ) whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor no...

Spectral Theory of Operator Measures in Hilbert Space

In §2 the spaces L2(Σ,H) are described; this is a solution of a problem posed by M. G. Krĕın. In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Năımark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function NΣ is introduced for an arbitrary (non...

Hilbert Space-valued Mixingales