We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded $H^{\infty }$-calculus and is based on elementary analysis.

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.

Given a strongly continuous semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on a Banach space X with generator A and an element f ∈ D(A²) satisfying $\left|\right|S\left(t\right)f\left|\right|\le e^{-\omega t}\left|\right|f\left|\right|$ and $\left|\right|S\left(t\right)A²f\left|\right|\le e^{-\omega t}\left|\right|A²f\left|\right|$ for all t ≥ 0 and some ω > 0, we derive a Landau type inequality for ||Af|| in terms of ||f|| and ||A²f||. This inequality improves on the usual Landau inequality that holds in the case ω = 0.

A new characteristic property of the Mittag-Leffler function $E_{\alpha }\left(a{t}^{\alpha }\right)$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.