H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions.
In this paper, we investigate operators on Riesz algebras, which are continuous with respect to multiplicative modifications of order convergence and relatively uniform convergence. We also introduce and study mo-Lebesgue, mo-$KB$, and mo-Levi operators.
In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.
We calculate in this paper the norm of composition, multiplicative and product operator, generated by multiplicative and measurable argument transformation between two different ordinary Lebesgue-Riesz and Grand Lebesgue spaces. We set ourselves the aim to obtain the exact expression for the norm of the considered operators by means of building of appropriate examples.
We discuss an analogue of the First Szegő Limit Theorem for multiplicative Toeplitz operators and highlight the role of the multliplicative Følner condition in this topic.
In the present paper, we introduce and investigate the multiplicative order compact operators from vector lattices to $l$-algebras. A linear operator $T$ from a vector lattice $X$ to an $l$-algebra $E$ is said to be $\mathbb{omo}$-compact if every order bounded net $x_α$ in $X$ possesses a subnet $x_{α_β}$ such that $Tx_{α_β}\moc y$ for some $y\in E$. We also introduce and study $\mathbb{omo}$-$M$- and $\mathbb{omo}$-$L$-weakly compact operators from vector lattices to $l$-algebras.
A multiplicative Hankel operator is an operator with matrix representation $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is the generating sequence of $M(α)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(α) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(β) \in \mathcal{M}_0$, $$\|M(α) - N(β)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(α) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} …
In \cite{Mor}, Marcantognini and Morán obtained Koplienko-Neidhardt trace formula for pairs of contractions and pairs of maximal dissipative operators via multiplicative path. In this article, we prove the existence of higher-order spectral shift functions for pairs of contractions and pairs of maximal dissipative operators via multiplicative path by adapting the argument employed in \cite{Mor}.
In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ for $f: \mathbb{Q}^+ \to \mathbb{C}$. This operator is the multiplicative analogue of the classical Toeplitz operator, and as such we denote the mapping by $\mathscr{M}_f$. We show that for $1 \leq p \leq q \leq \infty$, if $f \in \ell^r(\mathbb{Q}^+)$, then $\mathscr{M}_f:\ell^p \to \ell^q$ is bounded where $\frac{1}{r} = 1 - \frac{1}{p} + \frac{1}{q} $. Moreover, for the cases when $p=1$ with any $q$, $p=q$, and $q=\infty$ …
We study the algebra $\mathscr{E}'(\mathbb{R}^d)$ equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1y_1,\dots,x_dy_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(Ω)$, $Ω$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators on open subsets of $\mathbb{R}_+^d$.
We study parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a $C_0-$semigroup and one linear bounded operator with Wick-type multiplication, all of them set in the infinite dimensional space framework of white noise analysis. We prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the stationary case when the time-derivative is equal to zero.
It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(ζ(w+s)-1)dw$ with respect to the basis $(n^{-s})_{n\ge 2}$; here $ζ(s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal H}^2_0$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator o…
We prove an Ulam type stability result for a non-associative version of the multiplicativity equation, that is, $T(xy)=Ψ(T(x),T(y))$, where $T$ is a unital bounded operator acting from an amenable Banach algebra to a dual Banach algebra.
We consider the problem of maximal regularity for non-autonomous Cauchy problems $u'(t) + B(t)A(t)u(t) + P(t)u(t) = f(t), u(0) = u_0$ and $u'(t) + A(t)B(t)u(t) + P(t)u(t) = f (t), u(0) = u_0$. In both cases, the time dependent operators $A(t)$ are associated with a family of sesquilinear forms and the multiplicative left or right perturbations $B(t)$ as well as the additive perturbation $P(t)$ are families of bounded operators on the considered Hilbert space. We prove maximal $L_p$-regularity results and other regularity properties for the solutions of the previous problems under minimal regul…
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to …
Given a II$_1$-factor $\mathcal{M}$ with tracial state $τ$ and given an $\mathcal{M}$-bimodule $\mathcal{E}(\mathcal{M},τ)$ of operators affiliated to $\mathcal{M}$ and a trace $φ$ on $\mathcal{E}(\mathcal{M},τ)$, (namely, a linear functional that is invariant under unitary conjugation), we prove that $\det_φ:\mathcal{E}_{\log}(\mathcal{M},τ)\to[0,\infty)$ defined by $\det_φ(T)=\exp(φ(\log |T|))$ is a multiplicative map on the set $\mathcal{E}_{\log}(\mathcal{M},τ)$ of all affiliated operators $T$ such that $\log_+(|T|)\in\mathcal{E}(\mathcal{M},τ)$. Finally, we show that all multiplicative ma…
