Spectral theory for operator matrices related to models in mechanics

We derive various properties of the operator matrix where A ₀ is a uniformly positive operator and A ₀ ^?1/2 DA ₀ ^?1/2 is a bounded nonnegative operator in a Hilbert space H. Such operator matrices are associated with second-order problems of the form $ \ddot z(t) + A_0 z(t) + D\dot z(t) = 0 $ , which are used as models for transverse motions of thin beams in the presence of damping.     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra $\mathfrak{A}$ of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra $L_M^n (\Gamma )$ in terms of the symbol of this operator.     

Quasi-Banach operator ideals with a very strange trace

As shown by S. Lord, F. Sukochev, and D. Zanin (see [7]), the theory of singular traces is well understood for operators on the Hilbert space. The situation turns out to be completely different in the Banach space setting. Indeed, quite strange phenomena may occur. We will construct quasi-Banach operator ideals ${\mathfrak A}$ A with seemingly contradictory properties: On the one hand, ${\mathfrak A}$ A supports a continuous trace τ that vanishes at all finite rank operators, which means that τ is singular. On the other hand, ${\mathfrak A}$ A contains the identity map I _Z of an infinite-dimensional Banach space Z and τ (I _Z ) =  1. This implies that there exist operators ${T \in \mathfrak A (Z)}$ T ∈ A ( Z ) such that ${\tau (T^n) = 1}$ τ ( T n ) = 1 for ${n = 1,2,{\dots} \;}$ n = 1 , 2 , ? , which is impossible for singular traces in the case of a Hilbert space. As most counterexamples, the new operator ideals have no useful application. They provide, however, a deeper insight into the philosophy of traces.     

Dynamical system with boundary control associated with a symmetric semibounded operator

Let L ₀ be a closed densely defined symmetric semibounded operator with nonzero defect indices in a separable Hilbert space $\mathcal H$ . It determines a Green system $\{{\mathcal H}, {\mathcal B}; L_0, \Gamma_1, \Gamma_2\}$ , where ${\mathcal B}$ is a Hilbert space, and the $\Gamma_i: {\mathcal H} \to \mathcal B$ are operators connected by the Green formula $$ (L_0^*u, v)_{\mathcal H}-(u,L_0^*v)_{\mathcal H} =(\Gamma_1 u, \Gamma_2 v)_{\mathcal B} - (\Gamma_2 u, \Gamma_1 v)_{\mathcal B}. $$ The boundary space $\mathcal B$ and the boundary operators Γ _i are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC): $$ \begin{array}{lll} && u_{tt}+L_0^*u = 0, \quad u(t) \in {\mathcal H}, \quad t>0,\\ && u\big|_{t=0}=u_t\big|_{t=0}=0, \\ && \Gamma_1 u = f, \quad f(t) \in {\mathcal B},\quad t \geq 0. \end{array} $$ We show that this system is controllable if and only if the operator L ₀ is completely non-self-adjoint. A version of the notion of wave spectrum of L ₀ is introduced. It is a topological space determined by L ₀ and constructed from reachable sets of the DSBC. Bibliography: 15 titles.     

Endpoint Bounds for the Quartile Operator

It is a result by Lacey and Thiele (Ann. of Math. (2) 146(3):693–724, 1997; ibid. 149(2):475–496, 1999) that the bilinear Hilbert transform maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{p_{3}}(\mathbb{R})$ whenever (p ₁,p ₂,p ₃) is a Hölder tuple with p ₁,p ₂>1 and $p_{3}>\frac{2}{3}$ . We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when $p_{3}=\frac{2}{3}$ . We show that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ when p ₁,p ₂>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps $L^{p_{1}}(\mathbb{R}) \times L^{p_{2},\frac{2}{3}}(\mathbb{R}) $ into $L^{\frac{2}{3},\infty}(\mathbb{R})$ . We also provide weak type estimates and boundedness on Orlicz-Lorentz spaces near p ₁=1,p ₂=2 which improve, in the Walsh case, the results of Bilyk and Grafakos (J. Geom. Anal. 16 (4):563–584, 2006) and Carro et al. (J. Math. Anal. Appl. 357(2):479–497, 2009). Our main tool is the multi-frequency Calderón-Zygmund decomposition from (Nazarov et al. in Math. Res. Lett. 17(3):529–545, 2010).     

Sine, cosine transforms and classical function classes

We study the continuity and smoothness properties of functions f ∈ L ¹([0, ∞)) whose sine transforms $ \hat f_s $ and cosine tranforms $ \hat f_c $ belong to L ¹([0,∞)). We give best possible sufficient conditions in terms of $ \hat f_s $ and $ \hat f_c $ to ensure that f belongs to one of the Lipschitz classes Lip α and lip α for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg α and zyg α for some 0 < α ≤ 2. The conditions given by us are not only sufficient, but also necessary in the case when the sine and cosine transforms are nonnegative. Our theorems are extensions of the corresponding theorems by Boas from sine and cosine series to sine and cosine transforms.     

A Note on Quasi-paranormal Operators

Let T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted ${{\mathcal{QP}}}$ , of operators satisfying ${{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}$ for all ${{x \in \mathcal{H}}}$ . This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ₀ of the spectrum of ${{T \in \mathcal{QP}}}$ , then E is self-adjoint if and only if ${{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}$ .     

Stability of Nonlinear Subdivision and Multiscale Transforms

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms $\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$ in ? _∞(?) given by a stationary recursion of the form $v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$ involving a nonlinear subdivision operator S acting on ? _∞(?), the approach is extendable to other nonlinear multiscale transforms and norms, as well.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

This paper is concerned with estimations of solutions of the Sturm–Liouville equation $$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$ ( p ( x ) y ' ( x ) ) ' + ( μ 2 - 2 i μ d ( x ) - q ( x ) ) ρ ( x ) y ( x ) = 0 , x ∈ [ 0 , 1 ] , where ${\mu\in\mathbb{C}}$ μ ∈ C is a spectral parameter. We assume that the strictly positive function ${\rho\in L_{\infty}[0,1]}$ ρ ∈ L ∞ [ 0 , 1 ] is of bounded variation, ${p\in W^1_1[0,1]}$ p ∈ W 1 1 [ 0 , 1 ] is also strictly positive, while ${d\in L_1[0,1]}$ d ∈ L 1 [ 0 , 1 ] and ${q\in L_1[0,1]}$ q ∈ L 1 [ 0 , 1 ] are real functions. The main result states that for any r > 0 there exists a constant c _r such that for any solution y of the Sturm–Liouville equation with μ satisfying ${|{\rm Im}\, \mu|\leq r}$ | Im μ | ≤ r , the inequality ${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$ ∥ y ( · , μ ) ∥ C ≤ c r ∥ y ( · , μ ) ∥ L 1 is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C ₀-semigroup.     

Toeplitz Operators with BMO Symbols on Holomorphic Besov Spaces

We study the Toeplitz operator $T^{\beta }_{\mu }$ , on the holomorphic Besov spaces $B^p_s$ in the unit ball, for complex measures $\mu $ on the unit ball. We give sufficient conditions for which $T^{\beta }_{\mu }$ is bounded. In the case of positive measures or $\textit{BMO}^{\beta }$ symbols, we obtain necessary and sufficient conditions in terms of (weighted) Berezin transform and Carleson measures for Besov spaces.     

Generalized Whittle–Matérn and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator (???Δ) ^m and the Whittle–Matérn kernels related to the differential operator (???Δ?+?I) ^m . This is done by allowing general differential operators of the form $\prod_{j=1}^m(-\Delta+\kappa_j^2I)$ with nonzero κ _j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle–Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to $W_2^m(\ensuremath{\mathbb{R}}^d)$ . On the side, we prove that generalized inverse multiquadric kernels of the form $\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}$ are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle–Matérn form with a variable scale κ(r) between κ ₁,...,κ _m . We also consider the case where some of the κ _j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle–Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration.     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

Existence and compactness for the {\overline{\partial}} -Neumann operator on q-convex domains

The aim of this paper is to give a sufficient condition for existence and compactness of the \({\overline{\partial}}\) -Neumann operator N _q on \({L^2_{(0,q)}(\Omega)}\) in the case Ω is an arbitrary q-convex domain in \({\mathbb{C}^n}\) .     

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.     

The Eigenfunctions of the Hilbert Matrix

For each noninteger complex number ??, the Hilbert matrix $$H_\lambda= \biggl( \frac{1}{n+m+\lambda} \biggr)_{n,m\geq0}$$ defines a bounded linear operator on the Hardy spaces $\mathcal{H}^{p}$ , 1<p<??, and on the Korenblum spaces $\mathcal{A}^{-\tau}$ , ??>0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex ?? results by Hill and Rosenblum for real ??. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.     

The long time error analysis in the second order difference type method of an evolutionary integral equation with completely monotonic kernel

We study convergence of the numerical methods in which the second order difference type method is combined with order two convolution quadrature for approximating the integral term of the evolutionary integral equation $$ u^{\prime}(t)+\int_{0}^{t}\beta(t-s)\,A\,u\,(s)\;ds \, =\,0 ,~~~~ t>0,~~u(0)=u_{0}, $$ which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space H and \(\beta (t)\) is completely monotonic and locally integrable, but not constant. We establish the convergence properties of the discretization in time in the \(l_{t}^{1}(0,\infty ;\,H)\) or \(l_{t}^{\infty }(0,\infty ;\,H)\) norm.     

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$\begin{aligned} H^{+}(f(t)e^{i\omega t})(x)=-\!\!\!\!\!\!\int \nolimits _{\!\!\!0}^{\infty }e^{i\omega t}\frac{f(t)}{t-x}\,dt,\quad \omega >0,\quad x\ge 0, \end{aligned}$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$ , the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega $ are derived for each fixed $x\ge 0$ , which clarify the large $\omega $ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$ , we classify our discussion into three regimes, namely, $x=\mathcal O (1)$ or $x\gg 1$ , $0<x\ll 1$ and $x=0$ . Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega $ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.