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Let H be a fixed Hilbert space and B(H, H) be the Banach space of bounded linear operators from H to H with the uniform operator topology. Oscillation criteria are obtained for the operator differential equationwhere the coefficients A, C are linear operators from B(H, H) to B(H, H), for each t ≤ 0. A solution Y: R+ → B(H, H) is said to be oscillatory if there exists a sequence of points ti ∈ R+, so that ti → ∞ as i → ∞, and Y(ti) fails to have a bounded inverse. The main theorem states that a solution Y is oscillatory if an associated scalar differential equation is oscillatory.

On a Banach space the numerical range of an unbounded operator has a certain density property in the scalar field. Consequently all hermitian and dissipative operators are bounded. For a smooth or separable or reflexive Banach space the numerical range of an unbounded operator is dense in the scalar field.

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the typewhere A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the typeMlak (9) obtained some results concerning the limitations of solutions of the latter equation.

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

Kato [3] has introduced a class of operators called strictly singular operators. These operators have many properties in common with compact operators. In fact the concept of a strictly singular operator is an extension of the concept of a compact operator. Kato has proved that if X and X′ are Banach Spaces, then the singular operators of X into X′ forms a closed subspace of the space of bounded linear operators of X into X′ and if X = X′, then these operators forms a two-sided ideal in the ring of bounded linear operators on X. He has also shown that the Riers-Schauder theorem holds for the spectrum of a strictly singular operator. Gohberg, Feldman and Markus [23] have treated the same class of operators with an equivalent definition.

In Part I of this paper we shall be concerned with the representation as convolutions of continuous linear operators which act on various function- spaces linked with a locally compact group and which commute with left — or right — translations; cf. the results in [12]. For completeness some known results are included whenever they follow from the general procedure. We have tried to follow simple general approaches as much as possible.

This paper comprises a number of applications of the results of Part I. We use essentially the same notation as in Part I with a few additions necessary for the problems at hand.

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

A corollary of the main theorem presented in this note is a generalisation of the well-known result that a self-adjoint square root of a positive self-adjoint compact linear map in a Hilbert space is itself a compact linear map. The method used here exploits the techniques developed recently in the study of k-set contractions ((1), (2)).