Control subspaces of minimal dimension. Elementary introduction. Discotheca

In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ: , where Cyc A=R∶R is a subspace of Χ, dim R<+∞. Spqn (AⁿR∶n?0)=χ. Always disc A ?μ_A=(the multiplicity of the spectrum of the operator (dim R∶R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ? disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties ofdisc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H²); disc S=2, (but μ_S=i); disc S _n ^* =n (butμ=1), where S_n=S⊕. ⊕S; disc S=2, (but μ_S=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_N=1 ? (the operator N is reductive). There are other results also, and also a list of unsolved problems.     

Control subspaces of minimal dimension and root vectors

We investigate the following characteristic of a linear operator A in a Banach space X: disc {inf(dim R′:R′?R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AⁿR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system \(\dot x = Ax + Bu\) with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

Stratonovich-Weyl correspondence for compact semisimple Lie groups

Let G be a compact Lie group, M a G-homogeneous space and π a unitary representation of G realized on a Hilbert space of functions on M. We give a general presentation of the Stratonovich-Weyl correspondence associated with π. In the case when G is a compact semisimple Lie group and π _λ an irreducible representation of G with highest weight λ, we study the Stratonovich-Weyl symbol of the derived operator d π _λ (X) for X in the Lie algebra of G and its behavior as λ goes to infinity.     

Coherent sequences and threads

The combinatorial principle □(λ) says that there is a coherent sequence of length λ that cannot be threaded. If λ=κ⁺, then the related principle κ_□ implies □(λ). Let κ?ℵ₂ and X⊆κ. Assume both □(κ) and κ_□ fail. Then there is an inner model N with a proper class of strong cardinals such that X∈N. If, in addition, κ?ℵ₀² and n<ω, then there is an inner model Mn(X) with n Woodin cardinals such that X∈Mn(X). In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality ⁺(ℵ₀²) implies Projective Determinacy.     

Random fields associated with multiple points of the Brownian motion

Let X_t be the Brownian motion in $\text{R}$^d. The random set Γ = {(t₁,…, t_n, z): X_tl = ··· = X_tn = z} in $\text{R}$^{d + n} is empty a.s. except in the following cases: (a) n = 1, d = 1, 2,…; (b) d = 2, n = 2, 3,…; (c) d = 3, n = 2. In each of these cases, a family of random measures M_λ concentrated on Γ is constructed (λ takes values in a certain class of measures on $\text{R}$^d). Measures M_λ characterize the time-space location of self-intersections for Brownian paths. If n = d = 1, then M_λ(dt, dz) = λ(dz) N_z(dt) where N₂ is the local time at z. In the case n = 2, the set Γ can be identified with the set of Brownian loops. The measure M_λ “explodes” on the diagonal {t₁ = t₂} and, to study small loops, a random distribution which regularizes M_λ is constructed.     

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

We prove that the number of limit cycles which bifurcate from a two-saddle loop of an analytic planar vector field X ₀ under an arbitrary finite-parameter analytic deformation X _λ , λ ∈ (? ^N , 0), is uniformly bounded with respect to λ.     

Characterizingω 1 and the long line by their topological elementary reflections

Given a topological space 〈X, T〉 ∈M, an elementary submodel of set theory, we defineX _Mto beX ∩M with the topology generated by {U ∩M : U ∈T ∩M}. We prove that it is undecidable whetherX _Mhomeomorphic toω ₁ impliesX =X _M,yet it is true in ZFC that ifX _Mis homeomorphic to the long line, thenX =X _M.The former result generalizes to other cardinals of uncountable confinality while the latter generalizes to connected, locally compact, locally hereditarily LindelöfT ₂ spaces.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

A local version of Gearhart’s theorem

Let {e ^tA: t ≥ 0} be a C₀—semigroup on the Hilbert space ?. If x ₀ ∈ ? is such that the local resolvent R(λ,A) x ₀ admits a bounded holomorphic extension to the open half plane {Reλ > 0}, then lim _t→∞‖e ^tA R(λ₀, A) x ₀‖ = 0 for each λ₀ ∈ ρ(A). This resuit is used to find mild spectral conditions which ensure the decay at infmity to zero of solutions of higher order abstract Cauchy problems.     

Limits of integrals involving almost periodic functions

Let Sp ? R⁺ be a discrete countable set, let {α_λ}_λ∈Sp be a sequence in l¹(Sp) and f(x) ? ∑_λ∈Spα_λsin(λx). f is an almost periodic odd function with {λ: ±λ ∈ Sp} as spectrum. We give some conditions about the set S so that $\int _1^{+\infty}\ f(x){\rm sin}(Rx){dx\over x}\rightarrow 0$ whenever R → +∞, R ∈ S.     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.     

Some class of analytic functions related to conic domains

For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.     

Operator valued weights in von Neumann algebras,I

An operator valued weight is a kind of generalized conditional expectation from a von Neumann algebra M to a sub von Neumann algebra N. If T is a n.f.s. (normal, faithful, semifinite) operator valued weight from M to N, and φ is a n.f.s. weight on N, then φ ° T defines a n.f.s. weight on M. Our main result is that σ_t^φ°T extends σ_t^φ, and that the map φ → φ ° T preserves cocycle Radon Nikodym derivatives.     

Nontrivial expansions of zero and absolutely representing systems

В статье дается описа ние общего вида абсол ютно сходящегося в локаль но выпуклом пространствеH разлож ения нуля по системе ?_Λ:={e(λ_k)} _k=1 ^∞ , где ?λ∈CMe(λ) = λe(λ)M-линейный непрерывны й оператор вH иλ _k ∈C. При дополнительных пред положениях выясняет ся связь между наличием вH такого разложения нуля по системе ?_Λ и тем, что ?_Λ я вляется абсолютно представл яющей системой вH.     

Multiplication operators on sobolev disk algebra

In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.     

Structure of modules over the stereotype algebra ℒ(X) of operators

It is well known that every module M over the algebra ?(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ? = E ? X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ? (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $E \circledast X \subseteq M \subseteq E \odot X.$ . As a corollary, we show that if X is a nuclear Fréchet space with a basis, then each Fréchet module M over the stereotype operator algebra ?(X) can be uniquely represented as the projective tensor product of X by some Fréchet space E, $M = E \widehat \otimes X$ .     

Commutator estimates in von Neumann algebras

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c ₀ such that, given any ? > 0, |[a, u _ε ]| ? (1 ? ε)|aε ? c ₀| for some unitary operator u _ε ∈ M. In particular, every derivation δ: M → I (where I is an ideal in M) is inner, and δ = δ _a for a ∈ I.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.