Bounded and compact vectorial Hankel operators

Extract Although examples drive this book, we first provide a whirlwind survey of the general concepts of operator theory. We do not expect the student to master these topics now since they are covered in future chapters. A Hilbert space H is a complex vector space endowed with an inner product ⟨x,y⟩ that defines a norm ‖x‖=⟨x,x⟩ with respect to which H is (Cauchy) complete. The inner product on a Hilbert space satisfies the Cauchy–Schwarz inequality |⟨x,y⟩|⩽‖x‖‖y‖ for all x,y∈H⁠. Examples of Hilbert spaces include Cn (complex Euclidean space), ℓ2 (the space of square-summable complex sequences), and L2[0,1] (the Lebesgue space of square-integrable, complex-valued functions on [0,1]⁠). Vectors x,y in a Hilbert space H are orthogonal if ⟨x,y⟩=0⁠. The dimension of a Hilbert space H is the cardinality of a maximal set of nonzero orthogonal vectors. This book is almost exclusively concerned with Hilbert spaces of countable dimension. Every such Hilbert space has an orthonormal basis (un)n=1∞⁠, a (possibly finite) maximal orthogonal set such that ⟨um,un⟩=δmn for all m,n⩾1⁠. With respect to an orthonormal basis (un)n=1∞⁠, each x∈H enjoys a generalized Fourier expansion x=∑n=1∞⟨x,un⟩un that satisfies Parseval’s identity ‖x‖2=∑n=1∞|⟨x,un⟩|2. A subspace (a norm-closed linear submanifold) of H is itself a Hilbert space with the operations inherited from H⁠. If (wn)n=1∞ is a (possibly finite) orthonormal basis for a subspace M of H and x∈H⁠, then PMx=∑n=1∞⟨x,wn⟩wn belongs to M and satisfies ‖x−PMx‖⩽‖x−y‖ for every y∈M⁠. In short, PMx is the unique closest vector to x in M⁠. Furthermore, PM defines a linear transformation on H whose range is M⁠. It is called the orthogonal projection of H onto M and it satisfies PM2=PM and ⟨PMx,y⟩=⟨x,PMy⟩ for all x,y∈H⁠. Let H and K be Hilbert spaces. A linear transformation A:H→K is bounded if ‖A‖=sup{‖Ax‖K:‖x‖H=1} is finite. Let B(H,K) denote the set of bounded linear operators from H to K⁠. We write B(H) for B(H,H)⁠. The quantity ‖A‖ is the norm of A. Since B(H) is closed under addition and scalar multiplication, it is a vector space. Furthermore, since ‖A+B‖⩽‖A‖+‖B‖ and ‖cA‖=|c|‖A‖ for all A,B∈B(H) and c∈C⁠, it follows that B(H) is a normed vector space. Endowed with this norm, B(H) is complete and thus forms a Banach space. Moreover, the composition AB belongs to B(H) and ‖AB‖⩽‖A‖‖B‖ for all A,B∈B(H)⁠. Therefore, B(H) is a Banach algebra. For each A∈B(H,K)⁠, there is a unique A*∈B(K,H) such that ⟨Ax,y⟩K=⟨x,A*y⟩H for all x∈H and y∈K⁠. The operator A* is the adjoint of A; it is the analogue of the conjugate transpose of a matrix. One can show that A↦A* is conjugate linear, that A**=A⁠, ‖A‖=‖A*‖⁠, and ‖A*A‖=‖A‖2⁠. This additional structure upgrades B(H) from a Banach algebra to a C*-algebra. One can exploit adjoints to obtain information about the kernel and range of an operator. For most of the operators A∈B(H) covered in this book, we give the matrix representation [A]=[⟨Auj,ui⟩]i,j=1∞ with respect to an orthonormal basis (un)n=1∞ for H⁠. This matrix representation [A] defines a bounded operator x↦[A]x on the Hilbert space ℓ2 of square summable sequences that is structurally identical to A. Schur’s theorem helps us determine which infinite matrices define bounded operators on ℓ2⁠. Many of the operators covered in this book, such as the Cesàro operator, the Volterra operator, weighted shifts, Toeplitz operators, and Hankel operators, have fascinating structured-matrix representations. An important class of operators is the compact operators. These are the A∈B(H) such that (Axn)n=1∞ has a convergent subsequence whenever (xn)n=1∞ is a bounded sequence in H⁠. Equivalently, an operator is compact if it takes each bounded set to one whose closure is compact. Each finite-rank operator is compact and every compact operator is the norm limit of finite-rank operators. The compact operators form a norm-closed, *-closed ideal within B(H)⁠. Some operators have a particularly close relationship with their adjoint. For example, the operator M on L2[0,1] defined by (Mf)(x)=xf(x) satisfies M*=M⁠. Such operators are selfadjoint. If µ is a positive finite compactly supported Borel measure on C⁠, then the operator N on L2(μ) defined by (Nf)(z)=zf(z) satisfies (N*f)(z)=z―f(z)⁠, and thus N*N=NN*⁠. Such operators are normal. The operator (Uf)(eiθ)=eiθf(eiθ) on L2(T) satisfies U*U=UU*=I⁠. Such operators are unitary. Unitary operators preserve the ambient structure of Hilbert spaces and can serve as a vehicle to relate A∈B(H) with B∈B(K)⁠. We say that A is unitarily equivalent to B if there is a unitary U∈B(H,K) such that UAU*=B⁠. Unitary equivalence is often used to identify seemingly complicated operators with relatively simple ones.

We study Hankel operators on the weighted Fock spaces Fp γ. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu’s characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Localization principle for compact Hankel operators

It is an old problem in operator theory whether a pair of norm one compact Hermitian operators with "small" (in norm) commutator can be "well" approximated by a commuting pair of Hermitian operators.We show that, for operators of rank not exceeding n, such approximants exist provided ||[A, B]||/n 1/2 is small.This improves a result of Pearcy and Shields and sheds some new light on the original question and its relationship to a few related ones.

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T(a) + H(b) and its adjoint are described. Fredholm properties of Toeplitz plus Hankel operatorsT(a)+H(b) with piecewise continuous generating functions a and b have been studied for many years. These operators are considered in various Banach and Hilbert spaces, and the results ob- tained show that the structure of the algebras generated by such operators is much more complicated than the structure of the algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients defined on closed smooth curves. Moreover, calculating the index of Toeplitz plus Hankel operators, one encounters even more difficult problems, and the difficulti es grow if one attempts to study invertibility or one-sided invertibility of such operators. It is worth noting that, in general, Fredholm operators T(a) +H(b) are not one-sided invertible. Nev- ertheless, a few works where one-sided invertibility was discussed, have appeared in literature recently. They are mainly concerned with two special cases of Toeplitz plus Hankel operators—viz. with the operators having the form M(a) := T(a) +H(a) or the form f

Let \({\vartheta}\) be a measure on the polydisc \({\mathbb{D}^n}\) which is the product of n regular Borel probability measures so that \({\vartheta([r,1)^n\times\mathbb{T}^n) >0 }\) for all 0 < r < 1. The Bergman space \({A^2_{\vartheta}}\) consists of all holomorphic functions that are square integrable with respect to \({\vartheta}\). In one dimension, it is well known that if f is continuous on the closed disc \({\overline{\mathbb{D}}}\), then the Hankel operator H f is compact on \({A^2_\vartheta}\). In this paper we show that for n ≥ 2 and f a continuous function on \({{\overline{\mathbb{D}}}^n}\), H f is compact on \({A^2_\vartheta}\) if and only if there is a decomposition f = h + g, where h belongs to \({A^2_\vartheta}\) and \({\lim_{z\to\partial\mathbb{D}^n}g(z)=0}\).

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cn and let f∈L2(D). For the Hankel operator Hf on the Bergman space A2(D), it is shown that the essential norm of Hf in L2(D) is comparable to the distance norm from Hf to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).

ABSTRACTWe prove that the Markov–Stieltjes transform is a bounded non compact Hankel operator on Hardy space with Hilbert matrix with respect to the standard Schauder basis of and a bounded non-compact operator on Lebesgue space for and obtain estimates for its norm in this spaces. It is shown that the Markov–Stieltjes transform on is unitary equivalent to the Markov–Stieltjes transform on . Inverse formulas and operational properties for this transform are obtained.

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by $$S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}$$ . In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space $$\mathcal {L}^2[0,2\pi ]$$ is a simple point set $$\{0\}$$ .

Abstract. Let $\Omega \subset \C$ be open,X a Banach space and $W\subset X^\prime$ . We show that every $\sigma (X,W)\mbox{-holomorphic function } f: \Omega \to X$ is holomorphic if and only if every $\sigma(X,W)\mbox{-bounded}$ set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that $\varphi \circ f$ is holomorphic for all $\varphi \in W$ , where W is a separating subspace of $X^\prime$ to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of $\Omega$ , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers).

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

We consider linear operator equations of the form where $K:{\cal X}\to{\cal Y}$ is a compact linear operator between Hilbert spaces ${\cal X} \hbox{ and } {\cal Y}.$ We assume y to be attainable, i.e., that problem (1) has a solution x† = K†y of minimal norm. Here K† denotes the (Moore‐Penrose) generalized inverse operator of K, which is unbounded when K is compact, with infinite dimensional range. Hence problem (1) is ill‐posed and has to be regularized in order to compute a numerical solution. We want to approximate the solution x† of the equation (1), when only an approximation yδ of y is available with ‖yδ − y‖ ≤ δ, where δ is called the noise level. Since K†yδ is not a good approximation of x†, we approximate x† with xδα := Rαyδ where {Rα} is a family of continuous operators depending on a parameter α. A classical example is the Tikhonov regularization defined by Rα = (K*K + αI)−1 K*, where I denotes the identity and K* the adjoint of K.Recently, new Tikhonov based regularization methods have been proposed in [1], [2] and [3], under the name of fractional Tikhonov, to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order.In this paper, we firstly provide a saturation result similar to the well‐known saturation result for Tikhonov regularization [6]: indeed, Tikhonov regularization under suitable a‐priori assumption and a‐priori choice rule, α = α(δ) ∼ c(δ)2/3, is of optimal order and the best possible convergence rate obtainable is ‖xδα−x†‖ = O(δ2/3). On the other hand, let R(K) be the range of K and let Q be the orthogonal projector onto $\overline{R(K)},$, if then x† = 0, as long as R(K) is not closed, and this shows how Tikhonov regularization for an ill‐posed problem with compact operator never yields a convergence rate which is faster than O(δ2/3), since it saturates at this rate. Such results motivated us to introduce the iterated versions of fractional Tikhonov methods in the same spirit of the iterated Tikhonov method. We prove that those iterated methods can overcome the afore‐mentioned saturation results.Afterwards, inspired by the works [4,5] we introduce the nonstationary variants of our iterated methods. Differently from the nonstationary iterated Tikhonov, we have two nonstationary sequences of parameters. In the noise free case, we give sufficient conditions on these sequences to guarantee the convergence providing also the corresponding convergence rates. In the noise case, we show the stability of the proposed iterative schemes proving that they are regularization methods. For the brief space we have at disposal, we will only cite some of the main results we obtained for one kind of fractional Tikhonov algorithms, that we call weighted Tikhonov. For the results about the other fractional Tikhonov algorithm analysis, proofs, numerical examples and full details, we invite the interested reader to look at the full paper [7]. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

We give a characterization of infinite-time admissible observation operators for the right-shift semigroup on L 2[0,∞). Our main result is that if A is the generator of this semigroup and C is the observation operator, mapping D(A) into the complex numbers, then C is infinite-time admissible if and only if for all s in the open right half-plane. We derive this using Fefferman's theorem on bounded mean oscillation and Hankel operators. This result solves a special case of a more general conjecture which says that the same equivalence is true for any strongly continuous semigroup acting on a Hilbert space. For normal semigroups the conjecture is known to be true and then it is equivalent to the Carleson measure theorem. We derive some related results and partial results concerning the case when the signals are not scalar but with values in a Hilbert space.

We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.