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Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by
Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$, assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.
In this paper, we study the principal spectral theory of age-structured models with random diffusion. First, we provide an equivalent characteristic for the principal eigenvalue, the strong maximum principle and a positive strict super-solution. Then, we use the result to investigate the effects of diffusion rate on the principal eigenvalue. Finally, we study how the principal eigenvalue affects the global dynamics of the KPP model and verify that the principal eigenvalue being zero is a critical value.
where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.
The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.
We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.
We establish Bohr inequalities for operator-valued functions, which can be viewed as analogues of a couple of interesting results from scalar-valued settings. Some results of this paper are motivated by the classical flavour of Bohr inequality, while others are based on a generalized concept of the Bohr radius problem.
For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.
In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by
${(e_{(k,j)}^*)_j}$
, for
$k\in\N$
, let
$Z=\ell^\infty(X_k:k\kin\N)$
be their l∞-sum, and let
$T:Z\to Z$
be a bounded linear operator with a large diagonal, i.e.,
Under which condition does the identity on Z factor through T? The purpose of this paper is to formulate general conditions for which the answer is positive.
Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.
In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e.
$\|T^n-P\|\to 0$
, here P is a projection. We have showed that T is uniformly P-ergodic if and only if
$\|T^n-P\|\leq C\beta^n$
,
$0<\beta<1$
. In this paper, we prove that such a β is characterized by the spectral radius of T − P. Moreover, we give Deoblin’s kind of conditions for the uniform P-ergodicity of Markov operators.
If A is a real
$2n \times 2n$
positive definite matrix, then there exists a symplectic matrix M such that
$M^TAM=\text {diag}(D, D),$
where D is a positive diagonal matrix with diagonal entries
$d_1(A)\leqslant \cdots \leqslant d_n(A).$
We prove a maxmin principle for
$d_k(A)$
akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality
$d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$
For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.
In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential
$V(q)$
of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.
We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,
For any
$\alpha \in \mathbb {R},$
we consider the weighted Taylor shift operators
$T_{\alpha }$
acting on the space of analytic functions in the unit disc given by
$T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$
We establish the optimal growth of frequently hypercyclic functions for
$T_\alpha $
in terms of
$L^p$
averages,
$1\leq p\leq +\infty $
. This allows us to highlight a critical exponent.
We first introduce the weighted averaged projection sequence in $\text{CAT}(\unicode[STIX]{x1D705})$ spaces and then we establish some inequalities for the weighted averaged projection sequence. Using the inequalities, we prove the asymptotic regularity and the $\unicode[STIX]{x1D6E5}$-convergence of the weighted averaged projection sequence. Furthermore, we prove the strong convergence of the sequence under certain regularity or compactness conditions on $\text{CAT}(\unicode[STIX]{x1D705})$ spaces.
We investigate the real space H of Hermitian matrices in
$M_n(\mathbb{C})$
with respect to norms on
$\mathbb{C}^n$
. For absolute norms, the general form of Hermitian matrices was essentially established by Schneider and Turner [Schneider and Turner, Linear and Multilinear Algebra (1973), 9–31]. Here, we offer a much shorter proof. For non-absolute norms, we begin an investigation of H by means of a series of examples, with particular reference to dimension and commutativity.
For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.
Let
$\Omega \subset \mathbb {R}^N$
,
$N\geq 2$
, be an open bounded connected set. We consider the fractional weighted eigenvalue problem
$(-\Delta )^s u =\lambda \rho u$
in
$\Omega $
with homogeneous Dirichlet boundary condition, where
$(-\Delta )^s$
,
$s\in (0,1)$
, is the fractional Laplacian operator,
$\lambda \in \mathbb {R}$
and
$ \rho \in L^\infty (\Omega )$
.
We study weak* continuity, convexity and Gâteaux differentiability of the map
$\rho \mapsto 1/\lambda _1(\rho )$
, where
$\lambda _1(\rho )$
is the first positive eigenvalue. Moreover, denoting by
$\mathcal {G}(\rho _0)$
the class of rearrangements of
$\rho _0$
, we prove the existence of a minimizer of
$\lambda _1(\rho )$
when
$\rho $
varies on
$\mathcal {G}(\rho _0)$
. Finally, we show that, if
$\Omega $
is Steiner symmetric, then every minimizer shares the same symmetry.
We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.