Abstract: 1 Banach Spaces.- 2 Banach Algebras.- 3 Geometry of Hilbert Space.- 4 Operators on Hilbert Space and C*-Algebras.- 5 Compact Operators, Fredholm Operators, and Index Theory.- 6 The Hardy Spaces.- 7 Toeplitz Operators.- References.

Abstract: The evolution problem 0∈du/dt+A(t)u(t),u(s)=x, where theA(t) are nonlinear operators acting in a Banach space, is studied. Evolution operators are constructed from theA(t) under various assumptions. Basic properties of these evolution operators are established and their relationship to the evolution equation is determined. The results obtained extend several known existence theorems and provide generalized solutions of the evolution equation in more general cases.

Abstract: The short distance behavior of field operator products is analyzed. It is shown that under certain conditions operator product expansions can be derived which give complete information on the short distance behavior and lead to the construction of composite field operators.

Abstract: A new formalism, using standard-basis matrix operators, is presented for the study of the collective excitations and the thermodynamic properties of an ensemble of identical interacting quantum-mechanical systems subsequently called "ions," each ion having discrete energy levels. A model Hamiltonian is formulated in terms of these operators. The Hamiltonian contains terms which express the interaction of the ion with the crystal field and the external fields as well as terms which arise from the mutual interaction of ions. Using the doubletime temperature-dependent Green's-function technique, the equations of motion of the Green's functions of standard-basis operators are developed in the random-phase decoupling approximation. It is demonstrated that the temperature-dependent correlation functions of standard-basis operators, which are obtained from associated spectral Green's functions, lead to a set of equations that can be solved for the occupation probabilities of the single-ion energy levels. Hence, one can calculate the thermal-average expectation value of any quantum-mechanical operator representing a microscopic observable of a single ion, or a pair of ions (correlations). An important feature of the standard-basis matrix-operator formalism is that the single-ion terms, such as crystal field, molecular field, or external fields, are always treated exactly in any Green's-function decoupling scheme. In contrast, Green's-function methods which use angular momentum operators often necessarily treat single-ion terms in a decoupling approximation. As an example, the general standard-basis operator theory is applied to the Heisenberg ferromagnet in the presence of uniaxial single-ion crystal-field anisotropy, which has received extensive theoretical attention previously, with widely varying results.

Abstract: : The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

Abstract: Various extensions and generalizations of Bernstein polynomials have been considered among others by Szasz [13], Meyer-Konig and Zeller [8], Cheney and Sharma [1], Jakimovski and Leviatan [4], Stancu [12], Pethe and Jain [11]. Bernstein polynomials are based on binomial and negative binomial distributions. Szasz and Mirakyan [9] have defined another operator with the help of the Poisson distribution. The operator has approximation properties similar to those of Bernstein operators. Meir and Sharma [7] and Jam and Pethe [3] deal with generalizations of Szasz-Mirakyan operator. As another generalization, we define in this paper a new operator with the help of a Poisson type distribution, consider its convergence properties and give its degree of approximation. The results for the Szasz-Mirakyan operator can easily be obtained from our operator as a particular case.

Abstract: The canonical splitting of the multiplicities of the unit tensor (Wigner) operators in U(3) was used in I to determine explicitly one Wigner operator in each (arbitrary) multiplicity set. The denominator function whose zeroes define the null space of this Wigner operator is presented in a new form from which the complete identification of the null space is made. Using the properties of the intertwining number of U(3), the null spaces of all the U(3) Wigner operators are determined, and it is demonstrated that the null spaces of the operators belonging to a multiplicity set are simply ordered by inclusion. The Wigner operator previously obtained from the canonical splitting is shown to be the one having the maximal null space for its multiplicity set.

Abstract: In this chapter we develop the theory of boundary value problems for partial differential equations of the parabolic type; this theory is in a way the analogue of the non-variational elliptic theory developed in Sections 1–8, of Chapter 21.

Abstract: This paper is a continuation of [1] where we began the study of intertwining analytic Toeplitz operators. Recall that X intertwines two operators A and B if XA = BX. Let H2 be the Hilbert space of analytic functions in the open unit disk D for which the functions fr(θ) = f(reiθ) are bounded in the L2 norm, and H∞ be the set of bounded functions in H2. For φ ∊ Hφ, T φ (or T φ(z)) is the analytic Toeplitz operator defined on H2 by the relation (T φ f)(z) = φ(z)f(z). For φ ∊ H∞, we shall denote {φ(z): |z| < 1} by Range (φ) or φ(D). Then where and σ(Tφ) = Closure(φ(D)) [1]. If φ ∊ H∞ maps D into D, then we define the composition operator C φ on H2 by the relation (C φ f) (z) = f(φ(z)). J. Ryff has shown [11, Theorem 1] that C φ , is a bounded linear operator on H2.

Abstract: Most explicit information on the eigenfunctions of a Laplace operator on a compact manifold comes from computations where a high degree of symmetry is present. In these cases, eigenspaces may be of large dimension, the zeros of the eigenfunctions are often critical points, and the eigenfunctions usually have degenerate critical points. However, these properties are all unstable under small perturbations of the metric, and are therefore rather misleading to one's intuition. From the point of view of differential topology, the best possible properties the eigenfunctions can have would be the following: Property A. The eigenspaces are one-dimensional. Property B. 0 is not a critical value of the eigenfunctions (so the zero or nodal set is a manifold of codimension 1). Property C. The eigenfunctions are Morse functions (they have nondegenerate critical points). These properties are true for a residual set of metrics on a manifold. In the same vein we also establish similiar generic properties for bifurcations, and discuss how this approach can be used to attack the problem of invariant properties of nth eigenfunctions. The idea for this work originated in some similiar work of J. Albert on eigenfunctions [3]. His methods show that the use of transversality arguments can be avoided. More general theorems and more detailed proofs will appear elsewhere [7]. The main technical tools used are the Sard-Smale theorem and the transversality theorems which follow from it, although all the proofs can be carried out directly by exhibiting open dense sets. A map/: H -• E between two Banach manifolds is Fredholm if Dfx: TX(H)-> Tfix)(E) has finite-dimensional kernel and cokernel, and its index is the difference in these two dimensions. A point y e E is a regular value for ƒ if x e f ~ (y) implies Dfx is onto. A set of second Baire category is a residual set, and we have used generic to describe a property which is true for a residual set.

Abstract: The objective of this paper is to generalize the results of Lax and Phillips (41 and Walker (61 to include elliptic partial differential operators of all orders whose coefficients approach constant values at infinity with a certain swiftness. An example is given of an elliptic operator having an infinite-dimen- sional null-space whose coefficients slowly approach constant limiting values. 1. Introduction. Let L2(R1; Ck) denote the usual Hilbert space of equiva- lence classes of Ck-valued functions on Rn whose absolute values are Lebesgue- square-integrable over Rn. Given a positive integer m, let H (Rn; Ck) denote m

Abstract: This paper is concerned with positive operators acting in a partially ordered Banach space, and provides extensions of various theorems which involve operators of this kind which are in addition supposed to be compact. It is shown that any continuous linear positive map T has a positive eigenvector provided the spectrum of T contains a point with modulus greater than the radius of the essential spectrum of T : this result contains the well-known theorem of Krein-Rutman for compact operators. Various results connected to the Krein-Rutman theorem in a natural way are provided for non-compact positive linear operators, some involving k -set contractions and another which utilizes the notion of a projectionally compact operator. Two fixed point theorems for nonlinear positive operators are obtained by the use of topological degree theory for k -set contractions.

Abstract: This article concerns two simple types of bounded operators with real spectrum on a Hilbert space H. The purpose of this note is to suggest an abstract algebraic characterization for these operators and to point out a rather unexpected connection between such algebraic considerations and the classical theory of ordinary differential equations. Now some definitions. A Jordan operator has the form S + N where S is selfadjoint, N = 0, and S commutes with N. A sub-Jordan operator is the restriction of a Jordan operator / to an invariant subspace of /. A coadjoint operator T satisfies e~l e = I+A,s + A7s for some operators A, and A2 or equivalently T* — 3 77* T + 3 r* r2 ir3 = o. The main results are Theorem A. An operator T is Jordan if and only if both T and T are coadjoint. Theorem B. // T is coadjoint, if T has a cyclic vector, and if críT) =[a,b], then T is unitarily equivalent to ' multiplication by x' on a weighted Sobolev space of order 1 which is supported on \\.a, b]. Theorem C. // T is coadjoint and satisfies additional technical assumptions, then T is a sub-Jordan operator. Let us discuss Theorem C. Its converse, every sub-Jordan operator is coadjoint, is easy to prove. The proof of Theorem C consists of using Theorem B to reduce Theorem C to a question about ordinary differential equations which can be solved by an exacting application of the Jacobi conjugate point theorem for Sturm-Liouville operators. The author suspects that Theorem C is itself related to the conjugate point theorem. Introduction. This article concerns the two simplest types of bounded operators with real spectrum on a Hilbert space H. The purpose of this article is to suggest an abstract algebraic characterization for these operators and to point out a rather unexpected connection between such algebraic considerations and the classical theory of ordinary differential equations. In particular, our Theorem II which gives an algebraic characterization of certain sub-Jordan operators (defined below) seems very closely related to the classical theorem asserting that a SturmLiouville operator defined on the interval [a, b] is positive definite if and only if there are no points conjugate to a in the interval. One appealing thing is that Received by the editors August 2, 1971. AMS 1969 subject classifications. Primary 4730, 4740, 3442; Secondary 4760.

Abstract: Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.

Abstract: The concept of the so-called generalized polynomial operators is considered and applied especially to systems described by certain types of nonlinear differential equations. A theorem concerning local invertibility of polynomial operators is given. By an example it is shown how this theorem can be used to prove the existence of solutions, to construct those solutions, and to find a region of BIBO stability of the aforementioned systems. The treatment is quite general, being based on functional analysis. In particular, it can be applied to the systems analyzed by using functional series of Volterra type.

Abstract: Let A be the generator of a (C0) contraction semigroup on a Banach space. Let B be a dissipative operator with densely defined adjoint. Assume that the inequality j1Bxjj5 IlAxll + blxll holds on the domain of A. Then the closure of A +B generates a (CO) contraction semigroup. Let A be the generator of a (C0) contraction semigroup on a Banach space X. Let B be a dissipative operator on X in the sense of Lumer and Phillips [3]. Assume that 9(B)D)-9(A). Since A is closed it follows that there are constants a, b 1. On the other hand, Wiist [4] recently showed that if A and B are symmetric operators on a Hilbert space with A selfadjoint then the validity of (1) with a= 1 implies that A +B is essentially selfadjoint, i.e., has selfadjoint closure. (Kato [2] had proved a slightly weaker result, starting from the analogue of (1) with norms replaced by their squares.) In this note we use a simplified version of Wiist's argument to extend the result to dissipative operators in a rather general Banach space setting. THEOREM. Let X be a Banach space. Let A and B be as above with ?9(B)D1-9(A). Assume that there is a constant b< J such that, for all xce9(A), (2) jIBxjl _ IIAxIl + b lKxll. Suppose also that the adjoint B* has a dense domain in X*. Then the closure of A +B is the generator of a (CO) semigroup. Received by the editors July 22, 1971. AMS 1970 subject class~fications. Primary 47A55, 47B44; Secondary 47D05.

Abstract: The Wilson expansion of the field operator productA1(x1)A2(x2) may be used to define composite operators which are local with respect to 1/2(x1+x2) and depend in addition on a vector η proportional to the distancex1−x2. It is proved that the composite operators are polynomials in η, for fixed η2 ≠ 0, and that their dependence on η2 only involves powers of η2 and lgη2.