Weyl type theorems for operators satisfying the single-valued extension property

Let T be a bounded linear operator acting on a Banach space X such that T or its adjoint T^∗ has the single-valued extension property. We prove that the spectral mapping theorem holds for the B-Weyl spectrum, and we show that generalized Browder's theorem holds for f(T) for every analytic function f defined on an open neighborhood U of σ(T). Moreover, we give necessary and sufficient conditions for such T to satisfy generalized Weyl's theorem. Some applications are also given.     

Positive integrable elements relative to a left Hilbert algebra

Let $\text{U}$ be an achieved left Hilbert algebra. Let $\text{η∈D}{}^{\mathrm{?}}$ be an element such that π′(η) is a positive operator. Then, following M. A. Rieffel, η is called integrable if sup{(η|e)e∈U and ee^?e²} < + ∞. It is shown that η is integrable if and only if there is an element ζ∈D^flat; such π′(ζ) is positive and ζ is a square root of η in an appropriate sense. This is shown to be a generalization of Godement's well known theorem on the existence of a convolution square root for a continuous square-integrable positive-definite function on a locally compact group. An “integral” and an “L¹-norm” are then defined on the linear span of the positive integrable elements and the completion of this space, denoted by L¹($\text{U}$), is shown to be the predual of $\text{l}$($\text{U}$). “Godement's theorem” is then used to investigate square-integrable representations of $\text{U}$.     

Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).     

An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R ^U = A ^U then the conclusion is that A is integral over R.     

Asymptotically or super linear cooperative elliptic systems in the whole space

Under the assumption that F is asymptotically or super linear as |U| → ∞ with U = (u, v) ∈ ?², we obtain the existence of ground state solutions of a class of cooperative elliptic systems in ? ^N by using a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. To the best of our knowledge, there is no result published concerning the systems in the whole space ? ^N .     

Characterization of solutions to the generalized Cauchy-Riemann system

For a generalized biaxially symmetric potential U on a semi-disk D⁺, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ?D⁺. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ?D⁺.     

A U(n) generalization of Ramanujan's 1Ψ1 summation

In this paper we derive a U(n) generalization of Ramanujan's ₁Ψ₁ summation directly from a recent U(n) multiple series refinement of the q-binomial theorem. We also obtain a new generalization of the Macdonald identities for A⁽¹⁾_l.     

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U₀=0, U₁=1, Un=PUn−1−QUn−2 (n?2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,…,7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,…,12, the only non-degenerate sequences where gcd(P,Q)=1 and Un(P,Q)=□, are given by U₈(1,−4)=²¹², U₈(4,−17)=⁶²⁰², and U₁₂(1,−1)=¹²².     

On the renewal measure for Gaussian sequences

A form for U(t), the expected number of times a Gaussian sequence falls below a level of t, is given in terms of the mean M(x) and the variance V²(x) functions. It is shown that under general conditions U(t) ∼ M⁽⁻¹⁾(t), t → ∞. Moreover, if M and V are regularly varying at infinity functions, then U(t) − M⁽⁻¹⁾(t) is also regularly varying at infinity. A renewal theorem for stationary Gaussian sequences is given, where it is shown that the asymptotic behavior of U(t) − t/μ is determined by the asymptotic behavior of V²(t)/t.     

An elementary proof of the Macdonald identities for A(1)l

In this paper it is shown that the Macdonald identities for A⁽¹⁾_l are a natural consequence of the recent multivariable generalization of classical basic hypergeometric series known as basic hypergeometric series in U(n). More precisely, a U(n) multiple series generalization of the q-binomial theorem is derived and used to generalize Cauchy's elegant proof of Jacobi's triple product identity and to give a direct, elementary proof of the Macdonald identities for A⁽¹⁾_l.     

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.     

A synopsis of Combinatorial Integral Geometry

A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? A → A is associative gives rise to an associated Lie algebra $\text{L}$(A), so any coalgebra C whose comultiplication Δ : C → C ? C is associative gives rise to an associated Lie coalgebra $\text{L}$^c(C). The assignment C ? $\text{L}$^c(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint U^c to $\text{L}$^c. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor $\text{L}$. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → $\text{L}$ o U of the adjoint functor pair U ? $\text{L}$ is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) $\text{L}$^c o U^c → 1 of the adjoint functor pair $\text{L}$^c ? U^c.Theorem. For any Lie coalgebra M, the natural map$\text{L}$^c(U^cM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras ₀E(U^cM) and ₀E(S^cM) associated to U^cM and to S^cM = U^c(TrivM) when U^c(M) and S^c(M) are given the Lie filtrations. [Just as U^c(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so S^c(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]     

The linear model revisited

Following Gerber (1982), the annual gain of an insurance concern, G_t, is modeled as an autoregressive moving average process. The surplus process is defined as U_t = u + Σ^t_k=₁G_k. The distribution of U_t is obtained. It is proved that the traditional limit theorems, i.e., the central limit theorem, the strong law of large numbers and the law of the iterated logarithm, hold for U_t as t → ∞. Under certain conditions, bounds are provided for the probability of non-ruin in a finite time interval. It is proved that, if E[G_t] > 0 as t → ∞, the probability of non-ruin in an infinite time interval is positive.     

Central limit theorems for empirical andU-processes of stationary mixing sequences

This paper gives sufficient conditions for the weak convergence to Gaussian processes of empirical processes andU-processes from stationary β mixing sequences indexed byV-C subgraph classes of functions. If the envelope function of theV-C subgraph class is inL _p for some 2<p<∞, we obtain a uniform central limit theorem for the empirical process under the β mixing condition $$k^{p/(p - 2)} (\log k)^{2(p - 1)/(p - 2)} \beta _k \to 0 as k \to \infty $$ In the case that the functions in theV-C subgraph class are uniformly bounded, we obtain uniform central limit theorems for the empirical process and theU-process, provided that the decay rate of the β mixing coefficient satisfies β _k =O(k ^?r ) for somer>1. These conditions are almost minimal.     

Chapter 1. Extension of the fundamental theorem of finite semigroups

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U₁, U₂, or U₃. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U₁'s, U₂'s, and groups (“buildable” meaning “dividing a wreath product of”).We show that up to division U₁'s can be moved to the right and U₂'s, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U₂, even by a wreath product. From this it is deduced that U₁'s and U₂'s do not affect group complexity, that any semigroup buildable from U₁'s, U₂'s, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U₁'s on the right, all U₂'s on the left, and a group in the middle. This last fact is handy for developing charactérizations.An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U₁'s, U₂'s, and the groups. In addition it is shown, for i = 1,2,3, that if the unit U_i does not divide a semigroup S, then S can be built using only groups and units not containing U_i. Thus, it can be deduced that any semigroup which does not contain U₃ must have group complexity either 0 or 1. This then establishes that indeed U₃ is the determinant of group complexity, since it is already proved that both U₁ and U₂ are transparent with regard to the group complexity function, and it is known that with U₃ (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R₁” (i.e., does not contain U₁, i.e., is buildable form U₂'s and groups), and commutes to the right over groups iff it does not contain U₂ (i.e., is buildable from groups and U₁'s). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on ?^∞(Z,U), with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on ?p(Z,U), regardless of p∈[1,∞], as the union of point spectra of its limit operators considered as acting on ?^∞(Z,U).     

Boundary smoothness properties of Lipα analytic functions

LetU be an open set andb ∈ bdy(U). Let 0 < α< 1. Let A(U) denote the space of Lipα functions that are analytic onU, and a(U) the subspace lipα ∩ A(U). The space a(U ∪b), consisting of the functions that are analytic nearb, is dense in a(U). Letk be a natural number. We say that a(U) admits ak-th order continuous point derivation (cpd) atb if the functionalf → f^{(k) (b)} is continuous on a(U ∪b), with respect to the Lipα norm.     

On the complete controllability of linear nonautonomous systems

We suggest a geometric approach to the controllability of nonautonomous linear control systems of the form $\dot x = A(t)x + B(t)u$ , x ?? ? ⁿ , u ?? U ? ? ^m , with conical control constraint set U and continuous matrices A(t) and B(t). We derive two new complete controllability criteria, the first of which is reduced to the analysis of the arrangement of the cones ??^?1(t)B(t)U in the state space of the system [ $\dot \Phi (t) = \left. {A(t)\Phi } \right|(t)$ , ??(0) = E] and the second is based on the existence of appropriate controls bringing zero back to zero. We prove a theorem on the approximation of the control constraint set U by cones with finitely many generators lying inside the cone U with the preservation of the complete controllability property. We present a number of examples illustrating some peculiarities in the evolution of controllability sets of nonautonomous linear systems.