Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property $$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$ for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$ ...

The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its ${}^{\infty }$-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, ${\mathscr{H}}_{Nb}\left(E\right)$, and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on ${\mathscr{H}}_{Nb}\left(E\right)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $\phi \left(D\right){\equiv }^t\phi$, φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on ${\mathscr{H}}_{Nb}\left(E\right)$ is PDE-preserving for a set ℙ ⊆...

We study analytic models of operators of class $C_{·0}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T\in C_{·0}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that ${\cong }_{\theta }=A²ₘ\left(⁎\right)\ominus \theta H²\left(\right)$ and $T\cong P_{{}_{\theta }}{M}_z{|}_{{}_{\theta }}$, where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their...

Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator $u{C}_{\phi }$ on H() is defined by $u{C}_{\phi }f\left(z\right)=u\left(z\right)f\left(\phi \left(z\right)\right)$. We investigate the boundedness and compactness of $u{C}_{\phi }$ induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^p$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X=H^p$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_h$. We give...

Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with $\left|\alpha \right|>{\beta }^{-1}$, where $\beta \equiv in{f}_{\left|\right|x\left|\right|=1}\left|\right|T*x\left|\right|>0$. In particular, every co-analytic, right-invertible T in () is supercyclic.

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition $$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\&lt;r}|x-y{|}^{2-n}v\left(y\right)dy=0$$ for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$ $$\frac{\left|u\right(x\left)\right|}{|x{|}^N}\le C{\int }_{{\mathbf{R}}^3}|x-y{|}^{-1}\frac{\left|\Delta u\right(y\left)\right|}{|y{|}^N}dy\text{for}\text{a.e.}x\text{in}{\mathbf{R}}^3.$$

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that $su{p}_{\left|z\right|<1}(1-|z\left|²\right)\left|\right|f^{\text{'}}\left(z\right)\left|\right|<\infty$. A sequence (Tₙ)ₙ of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and ℓ₁(Y) if the map ${\sum }_{n=0}^{\infty }xₙzⁿ\to \left(Tₙ\left(xₙ\right)\right)ₙ$ defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if $su{p}_k{\sum }_{n=2^k}^{2^{k+1}}\left|\right|Tₙ\left|\right|²<\infty$. Several results about Taylor coefficients...

The tempered fundamental group of a $p$-adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between $\left(p^{\prime }\right)$ versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log...

For a $C^1$-function $f$ on the unit ball $𝔹\subset {ℂ}^n$ we define the Bloch norm by ${\parallel f\parallel }_{𝔅}=sup\parallel \tilde{d}f\parallel ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that $${\parallel f\parallel }_{𝔅}=\underset{\genfrac{}{}{0pt}{}{z,w\in 𝔹}{z\ne w}}{sup}{(1-|z|}^2{)}^{1/2}{(1-|w|}^2{)}^{1/2}\frac{\left|f\right(z)-f(w\left)\right|}{|w-P_{w}z-s_w{Q}_{w}z|}.$$

It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces $E₀\left(exp\alpha i\right)\otimes ̂E_{\infty }\left(exp\beta j\right)$ and $E₀\left(exp\alpha ̃i\right)\otimes ̂E_{\infty }\left(exp\beta ̃j\right)$. This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

The inequality $ʃlog|\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge Clog\left(\sum \right|a_n{{|}^2)}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of $a_n$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2\pi i\phi }$.

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r\le d$) where $(r,\theta ,z)$ denotes the cylindrical co-ordinates in ${ℝ}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, 1) $\mathrm{dist}(0z,\partial A)=O\left(k^{1/2}\right)$; $\mathrm{diam}A=O(exp(-c_0{k}^{3/2}))$; 2) ${\left(k^{1/2}\Psi \right)}_{k\in \mathbb{N}}$ converges to a vortex cylinder...