Inequalities for exponentials in Banach algebras

A. Pryde

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On the joint spectral radius of commuting matrices

Rajendra Bhatia, Tirthankar Вhattacharyya (1995)

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For a commuting n-tuple of matrices we introduce the notion of a joint spectral radius with respect to the p-norm and prove a spectral radius formula.

On a theorem of Lebow and Mlak for several commuting operators

J. Janas (1983)

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On the joint spectral radius of a nilpotent Lie algebra of matrices

Enrico Boasso (1999)

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For a complex nilpotent finite-dimensional Lie algebra of matrices, and a Jordan-Hölder basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems

Sen Huang (1995)

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Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem $u^{(n)} (t) = A u (t)$ , t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) $u (t) = p (t) + A ʃ_{0}^{t} a (t - s) u (s) d s$ , t ∈ ℝ, where $a (\cdot) \in L_{l o c}^{1} (ℝ)$ and p(·) is a vector-valued polynomial of the form $p (t) = \sum_{j = 0}^{n} 1 / (j!) x_{j} t^{j}$ for some elements $x_{j} \in E$ . We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension...

On spectral representation for selfadjoint operators. Expansion in generalized eigenelements

Eberhard Gerlach (1965)

Annales de l'institut Fourier

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L’auteur reprend l’étude classique de la représentation spectrale d’un opérateur auto-adjoint $A$ dans un espace de Hilbert $ℋ$ . Il y ajoute des précisions nouvelles qui conduisent à la définition du projecteur infinitésimal $P^{) λ)}$ sur l’espace des vecteurs propres généralisés $ℋ^{(λ)}$ . Il obtient, par conséquent, des énoncés plus précis de bien des théorèmes classiques. Il introduit ensuite la notion de “ $A$ -expansibilité” d’un sous-ensemble $S \subset ℋ$ . Cette notion est appliquée à l’étude des espaces fonctionnels...

An approximation problem in $L^{p} ([0, 2 π])$ , 2 < p < ∞

Daniel Oberlin (1981)

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Functional calculi for pseudodifferential operators, III

J. Alvarez (1989)

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The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin

Paweł Głowacki (1998)

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Let A be a pseudodifferential operator on $ℝ^{N}$ whose Weyl symbol a is a strictly positive smooth function on $W = ℝ^{N} \times ℝ^{N}$ such that $| \partial^{α} a | \leq C_{α} a^{1 - ϱ}$ for some ϱ>0 and all |α|>0, $\partial^{α} a$ is bounded for large |α|, and $l i m_{w \to \infty} a (w) = \infty$ . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin. ...

Commutators associated to a subfactor and its relative commutants

Hsiang-Ping Huang (2002)

Annales de l’institut Fourier

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Let $N \subseteq M$ be an inclusion of $I I_{1}$ factors with finite Jones index. Then $M = (N^{'} \cap M) \oplus [N, M]$ as a vector space. Here $[N, M]$ denotes the vector space spanned by the commutators of the form $[a, b]$ where $a \in N, b \in M$ .

Initial value problem for the time dependent Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz (1997)

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Let L be the full laplacian on the Heisenberg group $ℍ^{n}$ of arbitrary dimension n. Then for $f \in L^{2} (ℍ^{n})$ such that ${(I - L)}^{s / 2} f \in L^{2} (ℍ^{n})$ , s > 3/4, for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t L} f (x) |}^{2} d x \leq C_{ϕ} ∥ f ∥_{W^{s}}^{2}$ . On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group $ℍ^{n}$ , then for every s < 1 there exists a sequence $f_{n} \in L^{2} (ℍ^{n})$ and $C_{n} > 0$ such that ${(I - L)}^{s / 2} f_{n} \in L^{2} (ℍ^{n})$ and for a $ϕ \in C_{c} (ℍ^{n})$ we have $ʃ_{ℍ^{n}} | ϕ (x) | s u p_{0 < t \leq 1} {| e^{(\sqrt - 1) t Δ} f_{n} (x) |}^{2} d x \geq C_{n} ∥ f_{n} ∥_{W^{s}}^{2}, l i m_{n \to \infty} C_{n} = + \infty$ .