arXiv subjects

Jan Rozendaal

Publications and source records attributed to Jan Rozendaal.

28 records · Page 2Linked to original sources

arXiv2018

Functional calculus for $C_{0}$-groups using (co)type

We study the functional calculus properties of generators of $C_{0}$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-iA$ generate a $C_{0}$-group on a Banach space $X$ with type $p\in[1,2]$ and cotype $q\in[2,\infty)$. Then $A$ has a bounded $\mathcal{H}^{\infty}$-calculus from $\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e. $f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. As a corollary of our main theorem, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded $\mathcal{H}^{\infty}$-calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for $C_{0}$-groups. We extend our results to $R$-bounded operator-valued calculi, and we give an application to the theory of rational approximation of $C_{0}$-groups.

math.FA↗

arXiv2018

Stability theory for semigroups using $(L^{p},L^{q})$ Fourier multipliers

We study polynomial and exponential stability for $C_{0}$-semigroups using the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.

math.FA↗

arXiv2018

Sharp growth rates for semigroups using resolvent bounds

We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an $L^{p}$-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

math.FA↗

arXiv2017

Disintegration of positive isometric group representations on $\mathrm{L}^p$-spaces

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $μ$. We factorize the integral with respect to $μ$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,μ)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,λ)$, where $λ$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,μ)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,μ^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,μ^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},μ^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,μ^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,μ^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-direct integrals of Banach spaces and representations that are developed extend those in the literature.

math.RT↗

arXiv2017

Fourier multiplier theorems involving type and cotype

In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for $T_m$ without any smoothness properties of $m$. Under smoothness conditions the boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant.

math.FA↗

arXiv2016

Fourier multiplier theorems on Besov spaces under type and cotype conditions

In this paper we consider Fourier multiplier operators between vector-valued Besov spaces with different integrability exponents $p$ and $q$, which depend on the type $p$ and cotype $q$ of the underlying Banach spaces. In a previous paper we considered $L^p$-$L^q$-multiplier theorems. In the current paper we show that in the Besov scale one can obtain results with optimal integrability exponents. Moreover, we derive a sharp result in the $L^p$-$L^q$-setting as well. We consider operator-valued multipliers without smoothness assumptions. The results are based on a Fourier multiplier theorem for functions with compact Fourier support. If the multiplier has smoothness properties then the boundedness of the multiplier operator extrapolates to other values of $p$ and $q$ for which $\frac1p - \frac1q$ remains constant.

math.FA↗

arXiv2016

Functional calculus on real interpolation spaces for generators of $C_{0}$-groups

We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded $H^{\infty}_{1}$-calculus on real interpolation spaces. Additional results are derived from this.

math.FA↗

arXiv2016

Operator Lipschitz functions on Banach spaces

Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q}$, for $p<q$ and $p=q=1$, and for $X=Y=\mathrm{c}_{0}$. We also obtain results for $p\geq q$. We also study the estimate above in the setting of Banach ideals in $\mathcal{L}(X,Y)$. The commutator estimates we derive hold for diagonalizable matrices with a constant independent of the size of the matrix.

math.FA↗

arXiv2013

Convergence of subdiagonal Padé approximations of $C_{0}$-semigroups

Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for $-A$ the generator of a uniformly bounded $C_{0}$-semigroup $T$ on a Banach space $X$, the sequence $(r_{n}(-tA))_{n \in\mathbb{N}}$ converges strongly to $T(t)$ on $\textrm{D}(A^α)$ for $α>\frac{1}{2}$. Local uniform convergence in $t$ and explicit convergence rates in $n$ are established. For specific classes of semigroups, such as bounded analytic or exponentially $γ$-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.

math.FA↗

arXiv2013

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a Hilbert space, then for each $τ>0$ the operator $A$ has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy $f(z)=O(e^{-τ\textrm{Re}(z)})$ as $|z|\rightarrow \infty$. The bound of this calculus grows at most logarithmically as $τ\searrow 0$. As a consequence, $f(A)$ is a bounded operator for each holomorphic function $f$ (on a right half-plane) with polynomial decay at $\infty$. Then we show that each semigroup generator has a so-called (strong) $m$-bounded calculus for all $m\in\mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called $γ$-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.

math.FA↗