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where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other things that if B,
$A>0,$
then
$\mathcal {D}( w,\mu ) $
is operator subadditive on
$(0,\infty ) $
, that is,
$$ \begin{align*} \mathcal{D}( w,\mu ) ( A) +\mathcal{D}( w,\mu) ( B) \geq \mathcal{D}( w,\mu )(A+B). \end{align*} $$
From this, we derive that if
$f:[0,\infty )\rightarrow \mathbb {R}$
is an operator monotone function on
$[0,\infty )$
, then the function
$[ f( t) -f( 0) ] t^{-1}$
is operator subadditive on
$( 0,\infty ) .$
Also, if
$f:[0,\infty )\rightarrow \mathbb {R}$
is an operator convex function on
$[0,\infty )$
, then the function
$[ f( t) -f( 0) -f_{+}^{\prime }( 0) t ] t^{-2}$
is operator subadditive on
$( 0,\infty ) .$
Given a sequence $\varrho =(r_n)_n\in [0,1)$ tending to $1$, we consider the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $\mathbb {D}$ such that for any compact set K included in the unit circle ${\mathbb {T}}$, different from ${\mathbb {T}}$, the set $\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space ${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in $H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${\mathcal {C}}(K)$ for any compact set $K\subset {\mathbb {T}}$ different from ${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:f\mapsto f(r_n \cdot )$ acting on $H(\mathbb {D})$. Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.
Using the $H^{\infty }$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^{2}(\Omega,\mathbb {C}\otimes \mathbb {H})$. The operators that we consider are of the type
where $\overline {\Omega }$ is the closure of either a bounded domain $\Omega$ with $C^{1}$ boundary, or an unbounded domain $\Omega$ in $\mathbb {R}^{3}$ with a sufficiently regular boundary, which satisfy the so-called property $(R)$ (see Definition 1.3), $e_1,\, e_2,\, e_3\in \mathbb {H}$ which are pairwise anticommuting imaginary units, $a_1,\,a_2,\, a_3: \overline {\Omega } \subset \mathbb {R}^{3}\to \mathbb {R}$ are the coefficients of $T$. In particular, it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_{\alpha }(T)$ for $\alpha \in (0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial _{x_l}^{m}$, do not commute among themselves. This kind of result is to be understood in the more general setting of the fractional diffusion problems. The method used to construct the fractional power of a quaternionic linear operator is a generalization of the method developed by Balakrishnan.
In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers
$p< q$
do the equalities
$T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$
,
$k=p, q$
, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if
$p$
is odd and
$q$
is even, and negative otherwise. The case
$(p,q)=(1,2)$
closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on
$C^*$
-algebras is also provided.
We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.
Let
$C_{\||.\||}$
be an ideal of compact operators with symmetric norm
$\||.\||$
. In this paper, we extend the van Hemmen–Ando norm inequality for arbitrary bounded operators as follows: if f is an operator monotone function on
$[0,\infty)$
and S and T are bounded operators in
$\mathbb{B}(\mathscr{H}\;\,)$
such that
${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a=\{z\in \mathbb{C} \ | \ {\rm{re}}(z)\geq a\}$
, then
We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.
This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order
$\exp\{N\Lambda\}$
for a suitable constant
$\Lambda > 0$
, the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as
$\exp\{{-}N\Lambda\}$
. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.
We show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say
$d\in \{1,2\}$
. This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as
where
$\boldsymbol {A}_0$
is a distinguished self-adjoint extension and
$\Theta $
is a self-adjoint linear relation in
$\mathbb {C}^d$
. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to
$\boldsymbol {A}_0$
, i.e., it belongs to
$\mathcal {H}_{-1}(\boldsymbol {A}_0)$
, with possible “infinite coupling.” A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations
$\Theta $
.
The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.
As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.
We establish a theory of noncommutative (NC) functions on a class of von Neumann algebras with a particular direct sum property, e.g., $B({\mathcal H})$. In contrast to the theory’s origins, we do not rely on appealing to results from the matricial case. We prove that the $k{\mathrm {th}}$ directional derivative of any NC function at a scalar point is a k-linear homogeneous polynomial in its directions. Consequences include the fact that NC functions defined on domains containing scalar points can be uniformly approximated by free polynomials as well as realization formulas for NC functions bounded on particular sets, e.g., the NC polydisk and NC row ball.
Given a holomorphic self-map
$\varphi $
of
$\mathbb {D}$
(the open unit disc in
$\mathbb {C}$
), the composition operator
$C_{\varphi } f = f \circ \varphi $
,
$f \in H^2(\mathbb {\mathbb {D}})$
, defines a bounded linear operator on the Hardy space
$H^2(\mathbb {\mathbb {D}})$
. The model spaces are the backward shift-invariant closed subspaces of
$H^2(\mathbb {\mathbb {D}})$
, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.
For any
$r\in [0,1]$
we give an example of a rigid operator whose spectrum is the annulus
$\{\lambda\in \mathbb{C} : r \le |\lambda| \le 1 \} $
. In particular, when
$r=0$
this operator is rigid and non-invertible, and when
$r\in {\kern1pt}] 0,1 [ $
this operator is invertible but its inverse is not rigid. This answers two questions of Costakis, Manoussos and Parissis [Recurrent linear operators. Complex Anal. Oper. Theory8 (2014), 1601–1643].
We study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates
Let
$\mathcal {B}(\mathcal {H})$
be the algebra of all bounded linear operators on a complex Hilbert space
$\mathcal {H}$
. In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class
$H^{\infty }(\mathbb {D},\mathcal {B}(\mathcal {H}))$
of bounded analytic functions from the unit disk
$\mathbb {D}:=\{z \in \mathbb {C}:|z|<1\}$
into
$\mathcal {B}(\mathcal {H})$
. For the complete circular domain
$Q \subset \mathbb {C}^{n}$
, we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.
We show that
$L_1(L_p) (1 < p < \infty )$
is primary, meaning that whenever
$L_1(L_p) = E\oplus F$
, where E and F are closed subspaces of
$L_1(L_p)$
, then either E or F is isomorphic to
$L_1(L_p)$
. More generally, we show that
$L_1(X)$
is primary for a large class of rearrangement-invariant Banach function spaces.
We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.
In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.
We establish a Wold-type decomposition for isometric and isometric Nica-covariant representations of the odometer semigroup. These generalize the Wold-type decomposition for commuting pairs of isometries due to Popovici and for pairs of doubly commuting isometries due to Słociński.
We study super weakly compact operators through a quantitative method. We introduce a semi-norm
$\sigma (T)$
of an operator
$T:X\to Y$
, where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure
$\sigma (T)$
and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual
$T^*$
are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.