Admissible Observation Operators for the Right-Shift Semigroup

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

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.

An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces

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

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

Operators H H satisfying S ∗ H = H S {S^ \ast }H = HS where S S is a unilateral shift on Hilbert space are called Hankel operators. For a fixed shift S S of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.

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)

Quantum mechanics is one of the two fundamental pillar of modern physics. The success of the theory can be found everywhere in our everyday life and essentially in every new product that we build. We just have to remember that every semiconductor chip usually uses a quantum behavior in an essential way, for example quantum tunneling, to work. Until now, none of the thousand of experiments realized have succeeded to contradicted or to find a problem with the predictions given by quantum mechanics. However, in spite of this incredible success, many profound questions are still open. For example, we have some problems understanding the measurement, the coherence and the decoherence process, as well as the interpretation of what the theory tell us about the world we live in (Schlosshauer, 2005). Among the possible ways of investigation that we have, we think that stressing the foundations of the theory at the level of the mathematical structure, on which the theory stands, could be a good way to understand why and how the theory works. The mathematical structure of quantum mechanics consists in Hilbert spaces defined over the field of complex numbers (Birkhoff & Von Neumann, 1936). The success of the theory has led a number of investigators, over many decades, to look for general principles or arguments that would lead quite inescapably to the complex Hilbert space structure. It has been argued (Stueckelberg, 1960; Stueckelberg & Guenin, 1961), for instance, that the formulation of an uncertainty principle, heavily motivated by experiment, implies that a real Hilbert space can in fact be endowed with a complex structure. The proof, however, involves a number of additional hypotheses that may not be so directly connected with experiment. In fact Reichenbach (Reichenbach, 1944) has shown that a theory is not straightforwardly deduced from experiments, but rather arrived at by a process involving a good deal of instinctive inferences. This was also pointed out more recently by Penrose (Penrose, 2005, p. 59);

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. Introduction Weak type 2 and weak cotype 2 spaces were originally introduced and investigated by V.D. Milman and G. Pisier in [10] and weak Hilbert spaces by Pisier in [13]. A further detailed investigation can be found in Pisier’s book [14]. The first example of a weak Hilbert space which is not isomorphic to a Hilbert space is the 2-convexified Tsirelson space (called the convexified Tsirelson space in this paper). This follows from the results of W.B. Johnson in [4]. For a detailed study of the original Tsirelson space we refer to [3]. Let X be a Banach space with a symmetric basis. It was proved in [14] that if X is a weak Hilbert space, then it is isomorphic to a Hilbert space and this has lead to the belief that if X is just of weak cotype 2, then it is of cotype 2. However, this turns out not necessarily to be the case. The main result of this paper states that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. We now wish to discuss the arrangement of this paper in greater detail. In Section 1 we give some basic facts on properties related to weak type 2 and weak cotype 2 while Section 2 is devoted to a review of some results on the convexified Tsirelson space which we need for our main result. Most of these results are stated without proofs since they can be proved in a similar manner as the corresponding results for the original Tsirelson space. In Section 3 we make the construction of the symmetric convexified Tsirelson space, investigate its basic properties and prove our main result stated above. 02000 Mathematics Subject Classification: 46B03, 46B07. ∗Supported by NSF grants DMS 9706108 and 0102686. †Supported by the Danish Natural Science Research Council, grant 9801867.

The numerical range W(T) of a linear bounded operator T on a Hilbert space is the collection of complex numbers of the form (Tx,x) with x ranging over the unit vectors in the Hilbert space. In this study, firstly, the connection between numerical range of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the first investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical range of the operator A is in the form W(A)= co (⋃_{n=1}^{∞}W(A_{n}) ), where co (Ω), Ω⊂C denotes the convex hull of Ω. The numerical radius w(T) of a linear bounded operator T on a Hilbert space is a number which is given by w(T)=sup {|λ| : λ∈W(T) }. In this study, secondly, the connection between numerical radius of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the second investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical radius of the operator A is in the following form w(A)= sup w(A_{n}). The Crawford number c(T) of a linear bounded operator T on a Hilbert space is given by c(T)=inf {|λ| : λ∈W(T) }. In this study, thirdly, the connection between Crawford number of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the third investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), Re(A_{n})≥0 (≤0), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then Crawford number of the operator A is in the following form c(A)=inf c(A_{n}). Moreover, It is known that for a linear bounded accretive operator T if W(T)⊂{z ∈C:|argz| < ϕ} for any ϕ∈[0, π/ 2), then it is called a sectorial operator with vertex γ=0 and semi-angle ϕ. In this case, T ∈S_{ϕ}(H). In this study, finally, the connection between sectoriality of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the last investigation the result obtained is as follows: If for any n≥1,H_{n} is a Hilbert space, A_{n}∈L(H_{n}),A_{n}∈S_{ϕ_{n}}(H_{n}),H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n},A∈L(H), then for some ϕ∈[0,π/2), A∈S_{ϕ}(H) the necessary and sufficient condition is sup ϕ_{n}<ϕ. At the end of this study, an example including all these connections has been given.

Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see Sz\'ekely, Rizzo and Bakirov (2007) and Lyons (2013). It is known that the distance covariance, and its generalization $\alpha$-distance covariance, can be defined in several different ways that are equivalent under some moment conditions. The present paper considers four such definitions and find minimal moment conditions for each of them, together with some partial results when these conditions are not satisfied. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their ($\alpha$-)distance covariance is 0; this extends results by Lyons (2013) and Dehling et al. (2018+). The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Sz\'ekely, Rizzo and Bakirov (2007).

Let α, β γ, δ be complex numbers such that γδ ≠ 0. If A and B are bounded linear operators on the Hilbert space H such that γA + δB is right invertible then we study the operator norm of (αA + βB)(γA + δB)−1 using the angle φ between two subspaces ran A and ran B or the angle ψ = ψ(A, B) between two operators A and B where

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.