We study asymptotic behavior of $C_0$-semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)x\parallel =0$, ∀x ∈ X. If, moreover, f is a function in $L_{\alpha }^1\left({ℝ}_{+}\right)$ which is of spectral synthesis in a corresponding algebra $L_{{\alpha }_1}^1\left(ℝ\right)$ with respect to (iσ(A)) ∩ ℝ, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)f̂\left(T\right)\parallel =0$, where $f̂\left(T\right)={ʃ}_0^{\infty }f\left(t\right)T\left(t\right)dt$. Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of Katznelson-Tzafriri,...

Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Hölder spaces illustrates the abstract result.

This paper is devoted to the investigation of the abstract semilinear initial value problem du/dt + A(t)u = f(t,u), u(0) = u₀, in the "parabolic" case.

The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u\left(0\right)=x\in \overline{D\left(A\right)}$, with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B{(\lambda -A)}^{-1}x=\lambda {\int }_0^{\infty }{e}^{-\lambda t}V\left(s\right)xds$, where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0,\infty )\times \overline{D\left(A\right)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever...

We study a family of commuting selfadjoint operators $={\left(A_k\right)}_{k=1}^n$, which satisfy, together with the operators of the family $={\left(B_j\right)}_{j=1}^n$, semilinear relations ${⅀}_i{f}_{ij}\left(\right)B_j{g}_{ij}\left(\right)=h\left(\right)$, ($f_{ij}$, $g_{ij}$, $h_j:{ℝ}^n\to ℂ$ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra $U_q^{\text{'}}\left(so\left(3\right)\right)$.

Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes }}_{\tau }Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes }}_{\tau }Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with ${\tau }_p$ the associated $L^p$-tensor product...