Spectral analysis for infinite rank perturbations of unbounded diagonal operators

In this paper we study the spectral theory for the class of linear operators A _∞ defined on the so-called non-archimedean Hilbert space E _ω by, A _∞:= D + F _∞ where D is an unbounded diagonal linear operator and F _∞:= Σ _k=1 ^∞ u _k ? v _k is an operator of infinite rank on E _ω .     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Estimates for nonlinear harmonic measures on trees

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set D the value at the origin of the solution to u(x) = F((x, 0),...,(x,m ? 1)) for every x ∈ \(\mathbb{T}_m \) , a directed tree with m branches with initial datum f + χD. Here F is an averaging operator on ? ^m , x is a vertex of a directed tree \(\mathbb{T}_m \) with regular m-branching and (x, i) denotes a successor of that vertex for 0 ≤ i ≤ m ? 1.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator ${C_\varphi }$ , when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T _w ${C_\varphi }$ on the Hardy space H ², where w is a bounded measurable function on ?D which is continuous at each point of F(φ), φ ∈ S(2), and T _w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H ².     

Description of the algebraic structure of second-order differential operators that can be represented in divergence form

In the paper one proves that the second-order differential operators ?(x,u,u _x ,u _xx ), representable in divergence form, can be written as ?=c^AΔ_A, where Δ_A is the corresponding minor of order m of the Hessian \(\det (u_{xx} ) = \Delta _{\left( {\begin{array}{*{20}c} {1...n} \\ {1...n} \\ \end{array} } \right)} \) whose representation coefficients c^A=c^A(x,u,u _x ) satisfy certain algebraic relations. One introduces the concept of a second-order D-elliptic differential operator and one establishes that the totally elliptic operators of divergence form are linear with respect to the second derivatives of the function u.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Schauder theory for Dirichlet elliptic operators in divergence form

Let A be a strongly elliptic operator of order 2m in divergence form with Hölder continuous coefficients of exponent ${\sigma \in (0,1)}$ defined in a uniformly C ^1+σ domain Ω of ${\mathbb{R}^n}$ . Regarding A as an operator from the Hölder space of order m +  σ associated with the Dirichlet data to the Hölder space of order ?m +  σ, we show that the inverse (A ? λ)^?1 exists for λ in a suitable angular region of the complex plane and estimate its operator norms. As an application, we give a regularity theorem for elliptic equations.     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.     

Fractional BVPs with strong time singularities and the limit properties of their solutions

In the first part, we investigate the singular BVP \(\tfrac{d} {{dt}}^c D^\alpha u + (a/t)^c D^\alpha u = \mathcal{H}u\) , u(0) = A, u(1) = B, ^c D ^α u(t)| _t=0 = 0, where \(\mathcal{H}\) is a continuous operator, α ∈ (0, 1) and a < 0. Here, ^c D denotes the Caputo fractional derivative. The existence result is proved by the Leray-Schauder nonlinear alternative. The second part establishes the relations between solutions of the sequence of problems \(\tfrac{d} {{dt}}^c D^{\alpha _n } u + (a/t)^c D^{\alpha _n } u = f(t,u,^c D^{\beta _n } u)\) , u(0) = A, u(1) = B, \(\left. {^c D^{\alpha _n } u(t)} \right|_{t = 0} = 0\) where a < 0, 0 < β _n ≤ α _n < 1, lim _n→∞ β _n = 1, and solutions of u″+(a/t)u′ = f(t, u, u′) satisfying the boundary conditions u(0) = A, u(1) = B, u′(0) = 0.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on L _p (?₊) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L ₂(?₊) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.     

Bounds and positivity conditions for operator valued functions in a Hilbert lattice

In this paper we investigate regular functions of a bounded operator A acting in a Hilbert lattice and having the form A=D + T, where T is a positive operator and D is a selfadjoint operator whose resolution of the identity P(t) $(a\le s \le b)$ has the property $P(s_2)-P(s_1)\;\;(s_1<s_2)$ are non-negative in the sense of the order. Upper and lower bounds and positivity conditions for the considered operator valued functions are derived. Applications of the obtained estimates to functions of integral operators, partial integral operators, infinite matrices and differential equations are also discussed.     

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

We study the well-posedness of the second order degenerate integro-differential equations(P2):(Mu)(t)+α(Mu)(t) = Au(t)+ft-∞ a(ts)Au(s)ds + f(t),0t2π,with periodic boundary conditions M u(0)=Mu(2π),(Mu)(0) =(M u)(2π),in periodic Lebesgue-Bochner spaces Lp(T,X),periodic Besov spaces B s p,q(T,X) and periodic Triebel-Lizorkin spaces F s p,q(T,X),where A and M are closed linear operators on a Banach space X satisfying D(A) D(M),a∈L1(R+) and α is a scalar number.Using known operatorvalued Fourier multiplier theorems,we completely characterize the well-posedness of(P2) in the above three function spaces.     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian ?Δ _A u = ?divA(?u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset \({\Omega \subseteq \mathbb{R}^n}\) . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form $$\int_\Omega F_{\bar{A}}(|\xi|) \mu_1(dx) \leq \int_\Omega \bar{A}(|\nabla \xi|)\mu_2(dx),$$ where \({\bar{A}(t)}\) is a Young function related to A and satisfying Δ′-condition, while \({F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}\) . Examples involving \({\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}\) are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality ?Δ _p u ≥ Φ, leading to Hardy and Hardy–Poincaré inequalities with the best constants.     

An extremal problem related to Kolmogoroff’s inequality for bounded functions

LetA andB be positive numbers andm andn positive integers,m. Then there is for complex valued functions φ onR with sufficient differentiability and boundedness properties a representation wherev ₁ andv ₂ are bounded Borel measures withv ₁ absolutely continuous, such that there exists a function φ with ∣φ⁽ⁿ⁾∣ ?A and ∣φ∣ ?A onR and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.     

Homoclinic orbits for the second-order Hamiltonian systems with obstacle item

This paper is concerned with the existence of homoclinic orbits for the second-order Hamiltonian system with obstacle item, ü(t)-A u(t) =▽F (t, u), where F (t, u) is T-periodic in t with ▽F (t, u) = L(t)u + ▽R(t,u). By using a generalized linking theorem for strongly indefinite functionals, we prove the existence of homoclinic orbits for both the super-quadratic case and the asymptotically linear one.     

Lower bound and upper bound of operators on block weighted sequence spaces

Let $A = {({a_{n,k}})_{n,k \ge 1}}$ be a non-negative matrix. Denote by L _v,p,q,F (A) the supremum of those L that satisfy the inequality $$\parallel Ax{\parallel _{v,q,F}} \ge L\parallel x{\parallel _{v,p,F}}$$ where x ? 0 and x ε ? _p (v, F) and also v = (v _n ) _n=1 ^∞ is an increasing, non-negative sequence of real numbers. If p = q, we use L _v,p,F (A) instead of L _v,p,p,F (A). In this paper we obtain a Hardy type formula for L _v,p,q,F ( ${H_\mu }$ ), where ${H_\mu }$ is a Hausdorff matrix and 0 < q ? p ? 1. Another purpose of this paper is to establish a lower bound for ‖A _W ^NM ‖ _v,p,F , where A _W ^NM is the N?rlund matrix associated with the sequence W = {w _n } _n=1 ^t8 and 1 < p < ∞. Our results generalize some works of Bennett, Jameson and present authors.     

Abstract wave equations and associated Dirac-type operators

We discuss the unitary equivalence of generators G _A,R associated with abstract damped wave equations of the type ${\ddot{u} + R \dot{u} + A^*A u = 0}$ in some Hilbert space ${\mathcal{H}_1}$ and certain non-self-adjoint Dirac-type operators Q _A,R (away from the nullspace of the latter) in ${\mathcal{H}_1 \oplus \mathcal{H}_2}$ . The operator Q _A,R represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of A*A. In addition to the unitary equivalence results concerning G _A,R and Q _A,R , we provide a detailed study of the domain of the generator G _A,R , consider spectral properties of the underlying quadratic operator pencil ${M(z) = |A|^2 - iz R - z^2 I_{\mathcal{H}_1}, z\in\mathbb{C}}$ , derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where R represents an appropriate function of |A| is treated in depth, and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R?=?0) and damped (R ≠ 0) abstract wave equations as well as the cases ${A^* A \geq \varepsilon I_{\mathcal{H}_1}}$ for some ${\varepsilon > 0}$ and ${0 \in \sigma (A^* A)}$ (but 0 not an eigenvalue of A*A) are separately studied in detail.