Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

On the mean curvature of semi-Riemannian graphs in semi-Riemannian warped products

We investigate the mean curvature of semi-Riemannian graphs in the semi-Riemannian warped product M× _f ? _?? , where M is a semi-Riemannian manifold, ? _?? is the real line ? with metric ??dt ² (???=?±1), and f: M???^?+? is the warping function. We obtain an integral formula for mean curvature and some results dealing with estimates of mean curvature, among these results is a Heinz?CChern type inequality.     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts

Given an elliptic cusp form f and an automorphic form f?? on a definite quaternion algebra over ?, there is a theta lifting from (f, f??) to an automorphic form ?(f, f??) on the quaternion unitary group GSp(1, 1) generating quaternionic discrete series at the Archimedean place. The aim of this paper is to provide an explicit formula for Fourier coefficients of ?(f, f??) in terms of periods of f and f?? with respect to a unitary character ?? of an imaginary quadratic field. As an application, we show the existence of (f, f??) with ?(f, f??) ? 0.     

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

Lack of relative monotonicity among various measures of trihedral angles

We consider the various measures on trihedral angles that have appeared in the literature and we show that no two of these measures are monotone with respect to each other. In other words, for any measures f, g, there exist trihedral angles ??, ??, ??, ?? such that f(??) >? f(??), g(??) <? g(??), f(??) >? f(??), g(??) >? g(??). This is done through an elementary and systematic method based on multivariable calculus.     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

A note on shadowing with chain transitivity

Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map f_M on the space M(X) of probability measures on X, and a transformation f_K on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:

(a)

fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1.

(b)

fⁿ and (f_K)ⁿ are equicontinuous for all n ? 1.

In particular, we show that for a continuous map f : X → X of a compact metric space X with infinite elements, if f is a chain transitive map with the shadowing property, then fⁿ and (f_K)ⁿ are syndetically sensitive for all n ? 1. Also, we show that if f_M (resp. f_K) is chain transitive and syndetically sensitive, and f_M (resp. f_K) has the shadowing property, then f is sensitive.In addition, we introduce the notion of ergodical sensitivity and present a sufficient condition for a chain transitive system (X, f) (resp. (M(X), f_M)) to be ergodically sensitive. As an application, we show that for a L-hyperbolic homeomorphism f of a compact metric space X, if f has the AASP, then fⁿ is syndetically sensitive and multi-sensitive for all n ? 1.     

Riesz Transforms of Schr?dinger Operators on Manifolds

We consider Schr?dinger operators A=???+V on L ^p (M) where M is a complete Riemannian manifold of homogeneous type and V=V ⁺?V ^? is a signed potential. We study boundedness of Riesz transform type operators $\nabla A^{-\frac{1}{2}}$ and $|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$ on L ^p (M). When V ^? is strongly subcritical with constant ????(0,1) we prove that such operators are bounded on L ^p (M) for $p\in(p_{0}', 2]$ where $p_{0}'=1$ if N??2, and $p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$ if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ?A ^?1/2 and |V|^1/2 A ^?1/2 on L ^p (M) for p??(1,inf?(q ₁,N)) where q ₁ is such that $\nabla(-\Delta)^{-\frac{1}{2}}$ is bounded on L ^r (M) for r??[2,q ₁).     

The η invariant of the Atiyah–Patodi–Singer operator on compact flat manifolds

Let ${\mathcal{D}}$ be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the ?? invariant ???= ??(M) for any orientable compact flat manifold M. After giving an explicit expression for ??(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining ??(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p??? 7, ??(s) can be expressed as a multiple of L _??(s), an L-function associated to a quadratic character mod p, while ??(0) is a (non-zero) integral multiple of the class number h _?p of the number field ${\mathbb Q(\sqrt {-p})}$ . In the case of metacyclic groups of odd order pq, with p, q primes, we show that ??(0) is a rational multiple of h _?p .     

Invertible linear maps preserving {1}-inverses of matrices over PID

Let R be a PID,chR = 2,n > 1, M_n(R) be then xn full matrix algebra over R.f denotes any invertible linear map preserving {1}-inverses from M_n(R) to itself. In this paper, we have proven thatf is an invertible linear map on M_n(R) preserving {1}-inverses if and only iff satisfies any one of the following two conditions: (i) there exists a matrixP ? GL _n(R) such thatf(A) =PAP ^?1 for allA ? M _n(R), (ii) there exists a matrixP ? GL _n(R) such thatf(A) =PA ^t P^?1 forA ? M _n(R).     

On the packing dimension of the Julia set and the escaping set of an entire function

Let f be a transcendental entire function. We give conditions which imply that the Julia set and the escaping set of f have packing dimension 2. For example, this holds if there exists a positive constant c less than 1 such that the minimum modulus L(r, f) and the maximum modulus M(r, f) satisfy log L(r, f) ≤ c logM(r, f) for large r. The conditions are also satisfied if logM(2r, f) ≥ d logM(r, f) for some constant d greater than 1 and all large r.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Direct iterative methods for least-squares solutions to singular operator equations

Least-squares consistency and convergence of iterative schemes are investigated for singular operator equations (1) Tx = f, where T is a bounded linear operator from a Banach space to a Hilbert space. A direct splitting of T into T = M ? N is then used to obtain the iterative formula (2) x^(k+1) = M^?Nx^(k) + M^?f, where M^? is a least-squares generalized inverse of M. Cone monotonicity is used to investigate convergence of (2) to a least-squares solutions to (1), extending results given for the matrix case given by Berman and Plemmons.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications

We obtained new exact inequalities that estimate the L _?? -norm of the Riesz derivative D ^?? f of a function f defined on $ {\mathbb{R}^m} $ in terms of the uniform norm of the function itself and the L _s -norm of the function acted by the Laplace operator. On a class of functions f such that ||??f||s ?? 1, we solved the problem of approximation of an unbounded operator D ^?? by bounded ones and the problem of optimal recovery of the operator D ^?? on elements of this class given with known error.     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

On Elliptic Extensions in the Disk

Given two arbitrary functions f ⁽⁰⁾, f ⁽¹⁾ on the boundary of the unit disk D in \({\mathbb R}^2\), it is shown that there exists a second order uniformly elliptic operator L and a function v in L ^p , with L ^p second derivatives (1?p?Lv?=?0 a.e. in D and with v?=?f ⁽⁰⁾ and \(\frac{ \partial v}{\partial n} = f^{(1)}\) on \(\partial{D}\). A similar extension property was proved in Cavazzoni (2003) for any pair of functions f ⁽⁰⁾, f ⁽¹⁾ that are analytic; a result is obtained under weaker regularity assumptions, e.g. with \(\frac{\partial f^{(0)}}{\partial \theta}\) and f ⁽¹⁾ Hölder continuous with exponent \(\eta > \frac{1}{2}\).