A note on bi-contractive projections on spaces of vector valued continuous functions

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)

This paper is concerned with the existence and characterization of optimal growth paths in continuous time infinite horizon problems, and brings to bear on these issues mathematical techniques presented by the author in Chichilnisky (1977). These techniques, non-linear functional analysis in Hilbert (weighted L2 and Sobolev) spaces, permit generalization, clarification and simplification of existing results. In particular, the following points are of importance: (i) As Hilbert spaces are self-dual, continuous linear functionals on the commodity space are elements of the space, and thus have a totally natural interpretation as prices. We therefore obtain that commodity and price paths are in the same spaces, a useful property to prove existence in general equilibrium models. A failure to meet this condition has led to earlier work on characterization being troubled by the problem that supporting linear functionals are not always interpretable as prices, so that one cannot always give a price characterization of optimal and efficient paths. In the present framework this problem does not arise. (ii) A natural way to pose the problem of finding an optimal path is as one of maximizing a continuous function on a compact set. A major advantage of the techniques we use here is demonstrated by the fact that we are actually able to follow this approach. (iii) Because of (ii), it is possible to establish the existence of an optimal path without the use of any convexity assumptions on either the technology or the preferences. This is a major relaxation of earlier conditions, and enables existence results to be applied to models with increasing returns in production, non-convex preferences and endogenous population. For instance, in Lane (1977) both the feasible net production set and the preference ordering are in general non-convex due to the endogenous response of population growth to income. In Ryder and Heal (1973) and Wan (1970) non-convexities appear as well as in environmental and biological problems discussed for example in Clark (1976). In this latter case the logistic growth functions of biological populations introduces non-convexities in a natural way. (iv) It can be shown that the prices that arise within this framework provide a well-defined present value for every feasible program. This makes it possible to provide a full characterization of optimal and efficient paths in terms of profit maximization without any reference to transversality conditions. This characterization is thus analogous to that

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

QUASIHYPONORMAL θ-CLASS AND BN-CLASS COMPOSITION 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.

In this paper, we apply the resolvent operator theory and an approximating technique to derive the existence and controllability results for nonlocal impulsive neutral integro‐differential equations with finite delay in a Hilbert space. To establish the results, we take the impulsive functions as a continuous function only, and we assume that the nonlocal initial condition is Lipschitz continuous function in the first case and continuous functions only in the second case. The main tools applied in our analysis are semigroup theory, the resolvent operator theory, an approximating technique, and fixed‐point theorems. Finally, we illustrate the main results with the help of two examples.

We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

Let A be an operator with the polar decomposition A = U|A|. The Aluthge transform of the operator A, denoted by ?, is defined as ? = |A|1/2U |A|1/2. In this paper, first we generalize the definition of Aluthge transformfor non-negative continuous functions f,g such that f(x)g(x) = x (x ? 0). Then, by using this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if A is bounded linear operator on a complex Hilbert space H, then h (w(A)) ? 1/4||h(g2 (|A|)) + h(f2(|A|)|| + 1/2h (w(? f,g)), where f,g are non-negative continuous functions such that f(x)g(x) = x (x ? 0), h is a non-negative and non-decreasing convex function on [0,?) and ? f,g = f (|A|)Ug(|A|).

In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.

In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach theorem on extensions of linear functionals and the Banach-Steinhaus theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.

We investigate the evolution problem where H is a Hilbert space, A is a self-adjoint linear non-negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non-local non-linearities. Copyright © 1999 John Wiley & Sons, Ltd.

A capital letter means a bounded liner operator on a complex Hilbert space H , and B(H) denotes the algebra of all bounded linear operators on H equipped with operator norm ‖ · ‖ . An operator T is said to be positive (denoted by T 0) if (Tx,x) 0 for all x∈H and an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. Chaotic order is defined by logA logB for strictly positive operators A and B . A continuous real-valued function f defined on an interval J is called operator monotone if A B implies that f (A) f (B) for all self-adjoint operators A , B with spectra in J . The well known celebrated Lower-Heinz inequality asserts that if A B 0, then Aα Bα for any α ∈ [0,1] , which means that t → tα is operator monotone. Another well known example of operator monotone function is t → log t on (0,∞) . Recently, Furuta [5] showed that if A > B , then f (Aα ) > f (Bα) for all α ∈ (0,1] and if logA > logB , then there exists β ∈ (0,1] such that f (Aα) > f (Bα) for all α ∈ (0,β ] , where f (t) is a non-constant operator monotone function on [0,∞) . Comprehensive survey on related order preserving operator inequalities were given in [2,4]. In this paper, we will obtain an improvement which leads to the result of Moslehian and Najafi [6].

By virtue of Furuta inequality, we show some characterizations of the spectral order of positive operators on a Hilbert space from the viewpoint of the Kan- torovich type inequality and the Riccati equation. Let A and B be positive operators on a Hilbert space. Then the spectral order AB holds if and only if there exist a unique positive contraction Tp,u such that Tp,uA p+u 2 Tp,u = A up 4 B p A up 4 for all p ≥ 0 and u ≥ 0. This form interpolates the chaotic order, δ-order and the usual order continuously. 1 Introduction. An operator means a bounded linear operator on a Hilbert space H. An operator A is said to be positive ( denoted by A ≥ 0) if (Ax, x) ≥ 0 for all x ∈ H.I n particular, we denote by A> 0i fA is positive and invertible. The usual order A ≥ B for selfadjoint operators A and B is defined by A − B ≥ 0. Let A and B be positive invertible operators. We introduce a function order A ≥f B for a real valued continuous function f on (0, ∞) defined by A ≥f B ⇐⇒ f(A) ≥ f(B).

Stochastic electrodynamics is classical electrodynamics with the hypothesis of a random background radiation in the whole space. The probability distributions of the Fourier components of this electromagnetic radiation are assumed Gaussian. Lorentz invariance fixes the spectrum of the radiation except for a constant, measuring its intensity, which is identified with Planck’s constant. A formalism is developed to deal with general stochastic problems, associating a Hilbert space with every set of random variables. Then, a mapping is defined of the algebra of continuous functions of the random variables onto the algebra of operators on the Hilbert space. A correspondence is found between probability distributions of the random variables and vectors in the Hilbert space. The case of Gaussian random variables is considered in detail. The formalism is applied to the study of the radiation field when some known amount of radiation is present besides the random background. This closely resembles the usual quantization of the free radiation field, although there are significant differences. In particular, not all vectors of the Hilbert space represent physical states in the present theory.

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ⊂ R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A 2 R+⋯)−. It is shown that if the set {λ ∈ ℂ: dim ker (λ-A)* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skii. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If Mλ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then Mλ is cyclic, but disc Mλ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.