For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X,\left(A^{\alpha }\right))}_{\theta ,p}=x\in X|t^{-\theta Re\alpha }\psi \left(tA\right)\in L{⁎}^p((0,\infty );X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.

The paper generalizes the instruction, suggested by B. Sz.-Nagy and C. Foias, for operatorfunction induced by the Cauchy problem $$T_t:\left\{\begin{array}{c}t{h}^{\text{'}\text{'}}\left(t\right)+(1-t)h^{\text{'}}\left(t\right)+Ah\left(t\right)=0\hfill \\ h\left(0\right)=h_0\left(t{h}^{\text{'}}\right)\left(0\right)=h_1\hfill \end{array}\right.$$ A unitary dilatation for $T_t$ is constructed in the present paper. then a translational model for the family $T_t$ is presented using a model construction scheme, suggested by Zolotarev, V., [3]. Finally, we derive a discrete functional model of family $T_t$ and operator $A$ applying the Laguerre transform $$f\left(x\right)\to {\int }_0^{\infty }f\left(x\right)P_n\left(x\right)e^{-x}dx$$ where $P_n\left(x\right)$ are Laguerre polynomials [6, 7]. We show that the Laguerre transform...

Let $(L,|\left|·\right|{|}_L)$ be a Banach lattice of equivalence classes of real-valued measurable functions on a σ-finite measure space and $T=T\left(u\right):u=(u₁,...,u_d),u_i>0,1\le i\le d$ be a strongly continuous locally bounded d-dimensional semigroup of positive linear operators on L. Under suitable conditions on the Banach lattice L we prove a general differentiation theorem for locally bounded d-dimensional processes in L which are additive with respect to the semigroup T.

Let L be a Banach lattice of real-valued measurable functions on a σ-finite measure space and T=$T_t$: t < 0 be a strongly continuous semigroup of positive linear operators on the Banach lattice L. Under some suitable norm conditions on L we prove a general differentiation theorem for superadditive processes in L with respect to the semigroup T.

Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to define and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces....

The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^0$-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^0$, the $H^0$-limit of $A^{\epsilon }$, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar’s method...