We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin, Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.
We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.
The main result of this paper, Theorem 1.1, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on spaces of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.
We provide decompositions of Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of Hausdorff state spaces. Combining both formulae we write every Dirichlet form as the sum of a recurrent, dissipative and transient conservative Dirichlet forms. Besides, we prove that Mosco convergence preserves invariant sets and that a Dirichlet form shares the same invariants sets with its approximating Dirichlet forms E(t) and E(?). Finally we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the co…
The ergodic decomposition of a family of Hua-Pickrell measures on the space of infinite Hermitian matrices is studied. Firstly, we show that the ergodic components of Hua-Pickrell probability measures have no Gaussian factors, this extends a result of Alexei Borodin and Grigori Olshanski. Secondly, we show that the sequence of asymptotic eigenvalues of Hua-Pickrell random matrices is balanced in certain sense and has a "principal value" coincides with the $γ_1$ parameter of ergodic components. This allow us to complete the program of Borodin and Olshanski on the description of the ergodic deco…
The aim of this paper is to prove ergodic decomposition theorems for probability measures quasi-invariant under Borel actions of inductively compact groups (Theorem 1) as well as for sigma-finite invariant measures (Corollary 1). For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by the initial invariant measure (Theorem 2).
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.
Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes $\{s \in \mathbb{C} \ | \ {\rm Re} s > \varepsilon \}$ for each $\varepsilon>0$. The properties of the formal inverse of the differentiation are also investigated.
Classical and non classical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows to develop Besov and Triebel-Lizorki…
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_φ$ on $\mathscr{H}^2$ which are generated by symbols of the form $φ(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $φ$, then the operator $\mathscr{C}_φ$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_φ$. Furthermore, in the…
We provide a fairly large class of II$_1$ factors $N$ such that $M=N\bar{\otimes}R$ has a unique McDuff decomposition, up to isomorphism, where $R$ denotes the hyperfinite II$_1$ factor. This class includes all II$_1$ factors $N=L^{\infty}(X)\rtimesΓ$ associated to free ergodic probability measure preserving (p.m.p.) actions $Γ\curvearrowright (X,μ)$ such that either (a) $Γ$ is a free group, $\mathbb F_n$, for some $n\geq 2$, or (b) $Γ$ is a non-inner amenable group and the orbit equivalence relation of the action $Γ\curvearrowright (X,μ)$ satisfies a property introduced in \cite{JS85}. On the…
The behaviour of solutions to fourth order problems is studied through the decomposition into a system of second order ones, which leads to relaxed formulations with the introduction of measure terms. This allows to solve a shape optimization problem for a simply supported thin plate.
In Bergman and Dirichlet spaces, the shift operator is not an isometry, but it is a left invertible operator. In this paper we give conditions on the left invertible operators such that a operator version, in the sense of Rosenblum and Rovnyak, of the Wold decomposition to take place.
We give three characterizations of the Dirichlet-type spaces $D(μ)$. First we characterize $D(μ)$ in terms of a double integral and in terms of the mean oscillation in the Bergman metric, none of them involve the use of derivatives. Next, we obtain another characterization for $D(μ)$ in terms of higher order derivatives. Also, a decomposition theorem for $D(μ)$ is established.
It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pulé condensation of type III demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages.
We consider a quasi-regular Dirichlet form. We show that a bounded signed measure charges no set of zero capacity associated with the form if and only if the measure can be decomposed into the sum of an integrable function and a bounded linear functional on the domain of the form. The decomposition allows one to describe explicitly the set of bounded measures charging no sets of zero capacity for interesting classes of Dirichlet forms. By way of illustration, some examples are given.
We prove that any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $Γ$ with positive first $\ell^2$-Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.
For a topological dynamical system we characterize the decomposition of the state space induced by the fixed space of the corresponding Koopman operator. For this purpose, we introduce a hierarchy of generalized orbits and obtain the finest decomposition of the state space into absolutely Lyapunov stable sets. Analogously to the measure-preserving case, this yields that the system is topologically ergodic if and only if the fixed space of its Koopman operator is one-dimensional.
The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. Another tool is a Drazin-like inverse operator $B$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This oper…
The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of "corners" of finite size, must be finite. A similar result, Theorem 1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results, combined with the ergodic decomposition theorem of [3], imply that the infinite Hua-Pickrell measures of Borodin and Olshanski [2] have finite ergodic components…
