Unbounded Case Research Articles

The square root of a positive self-adjoint operator

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.

Self-consistent theory of bounded rotational spectra: (I). Two-dimensional rotator

The effect of an upper bound of the spectrum for rotational motion of a set of particles confined to move in a plane is studied by means of the generalized Hartree-Fock approximation (GHFA) without pairing. The results can be expressed as corrections to the wave-functions and the collective energies of the A-nucleon and ( A±1)-nucleon systems of the previously considered unbounded case. Full self-consistency of the basic equations in the bounded case is assured under the same restrictions that limited the general accuracy of the method for the unbounded spectrum. A detailed quantitative discussion of the collective energies of the A-nucleon system shows results that are interesting enough to warrant an investigation of the full three-dimensional problem.

A Note on Square Roots of Positive Operators

1. The study of the self-adjoint operators of mathematical physics by variational methods (as, for instance, in the monograph of Mikhlin [1]) utilizes Hilbert space theory in an elementary sense, in that no use is made of the spectral theorem for self-adjoint operators except at one point. Whenever the square root of a positive unbounded self-adjoint operator is required, the spectral theorem is utilized to establish the existence of the square root. Elementary proofs of the existence of the square root for positive self-adjoint bounded operators are well known (cf., e.g., [2] or [3]). We present here an elementary proof of the existence of the square root for the unbounded case. 2. Let H be a complex Hilbert space with inner product (.,.) and norm 11. A linear operator A with domain DA dense in H possesses an adjoint A* which is a closed linear operator; if A = A*, A is self-adjoint. If (Ax, x) > y2(x, x), Py > 0 for all x E DA, A will be called positive. If (Ax, x) > 0 for all x E DA, X # 0, then A will be called positive definite. If (Ax, x) _ 0 for all x E DA, A will be called positive. We write A ? B if A B is positive. An elementary proof exists for the following theorem: Every positive bounded self-adjoint operator A possesses a unique positive bounded self-adjoint square root which commutes with every bounded operator which commutes with A. We will utilize this result and prove first: THEOREM 1. Every positive self-adjoint operator A possesses a unique positive self-adjoint square root which commutes with every bounded operator that commutes with A. (Here A commutes with a bounded operator T if TA extends AT.) Next we will show: THEOREM 2. Theorem 1 remains true if the word strictly is removed. 3. Proof of Theorem 1. From Schwarz's inequality we see that

Theory of Signal Detectability: Deferred-Decision Theory

The theory of signal detectability is extended to include observation-decision procedures in which the available observation time is bounded. The special case of a simple signal hypothesis with stationary normal observation statistics is worked in detail. (“Signal known exactly in added white Gaussian noise” is an example of such a case.) The optimization is of the minimum average risk type, with constant cost of observation to facilitate comparison with work based on Wald's sequential analysis and comparison with fixed observation procedures. An unexpected result is that for large available observation lengths, approaching Wald's unbounded case, the optimization dictates that the primary improvement is in error performance rather than observation time.

Optimal Bounded Control of Linear Sampled-Data Systems With Quadratic Loss

A method is presented for the computation of optimal control for linear sampled-data systems when the control variable is a bounded scalar. It is shown that for this problem the optimal control is a piecewise linear function of the state and may be computed by piecewise iteration of suitable recurrence relations. The optimal control is presented in terms of the control coefficients (matrices) and the regions to which they apply. No solution other than computer storage is suggested for the synthesis of these controls. In the second section of the paper, it is shown that the method applies with trivial modification to the random-input, random-observation noise case. The optimal control law has the same form as the deterministic case with the conditional expectation used in the control law in place of the stale itself. A simple deterministic example computed on an IBM 1620 is presented. As might be expected, the computer capacity required for the problem is intermediate between the unbounded control case, where the control is linear, and more general problems.

Propagation and Instabilities of Waves in Bounded Finite Temperature Plasmas

The quasistatic analysis is used to examine propagation and instabilities of waves in a plasma in which the transverse bounds and the temperature are important. A dispersion equation is derived by use of a dielectric tensor, which is correct within the quasistatic assumption and the assumption of longitudinal velocity, only. The dispersion curves have a smooth transition from the zero temperature case at long wavelengths to the unbounded finite temperature case at short wavelengths. The stop-bands which appear in the zero temperature analysis become propagating regions in the more general case. Landau-type damping is calculated; the cyclotron wave is strongly damped. The effect of the boundaries and the magnetic field on a double-humped velocity distribution is examined. The transverse boundaries are stabilizing with respect to space-charge waves, but the cyclotron interactions are strengthened. The temperature always exerts a stabilizing influence which is particularly marked for the interaction between the cyclotron waves. Curves of the limiting regions of stability, with respect to various parameters, are given, illustrating these effects. By the use of an example it is demonstrated that a moderate temperature can suppress the cyclotron instability. This result is in agreement with the experimental observation that a plasma, predicted to be unstable with respect to the cyclotron-wave interaction, was actually stable.

On closed operators with closed range

It is well-known that if T is an everywhere defined bounded operator on a Banach space X to a Banach space Y and T* is its adjoint, then the range R(T*) of T* is closed in X* if and only if the range R(T) of T is closed in Y (cf. [2, pp. 487-489]). The object of this note is to establish this result for closed but possibly unbounded operators. This result, for the unbounded case, is of great utility in the study of differential operators and has been considered by F. E. Browder,2 I. C. Gohberg and M. G. Kreln, and by G. C. Rota. In the sequel, we shall have occasion to consider a set under several different topologies. We shall use the following convention: if A is a linear set and B is a set of linear functionals on A, then the set A with the weak topology induced by the elements of B will be denoted by (A, B). Thus, the assertion that a set C is closed (dense, etc.) in (A, B) shall mean that C is a subset of A which is closed (dense, etc.) in this weak topology. If A is a Banach space, then the assertion that a set C is closed (dense, etc.) in A shall refer to the norm topology of A. Henceforth, X and Y will be Banach spaces. If T is a closed operator with domain D(T) in X and range R(T) in Y, then it was noted by Sz.-Nagy [5], that D(T) becomes a Banach space under the norm I XT = I X I + ITxI (which we shall call the T-norm) and T is a bounded operator on this space. If in addition, D(T) is dense in X, it is well known that T has a uniquely defined adjoint T* with domain D(T*) dense in (Y*, Y), and that T* is a closed operator.

On the Permutability of Normal Operators

1. In this paper we obtain a theorem about the permutability of (in general unbounded) normal operators. This theorem is analogous to a theorem proved by the present authors in collaboration with J. von Neumann [3] concerning the permutability of self-adjoint operators. As we shall point out later it is not possible to generalize precisely the conditions given in [3] to the case of normal operators. Nevertheless, the proof of the theorem given here, with essential modifications, follows the proof given in [3]. We give our theorem and proof in Section 2. In Section 3 we apply this theorem on the permutability of normal operators to obtain the integral representation for a semi-group {N,} of normal operators (in general unbounded), where the index x ranges over a semi-group which is a cartesian product I+ X E where I+ is the set of vectors in m-dimensional euclidean space with non-negative integer components and Et is the set of vectors in n-dimensional euclidean space with non-negative components. This theorem contains as special cases theorems by E. Hille [6], B. v.Sz. Nagy [9, pp. 74-76], A. Devinatz [2] and a theorem recently published by R. Getoor [5]. It should be pointed out that our theorem represents an essential generalization of these previous results. For the case where { Nx ; x > 0 } is a one parameter semi-group of unbounded normal operators, the method of Nagy [9, pp. 74-75] may be pushed through to obtain the integral representation, and only in this case. However, with the use of our permutability theorem we are able to materially simplify the proof for the one parameter case. Moreover, in the unbounded case, even with the use of our permutability theorem, there remain some difficulties for multi-parameter semi-groups which do not allow us to use immediately the theorem for the one-parameter case. If the index x ranges over a more general Abelian topological semi-group than we consider, the representation problem is still open, although for semi-groups of normal operators which are uniformly bounded a method introduced by R. Phillips [8] may be used (cf. also A. E. Nussbaum [7]). 2. We start with the definition of permutability of two (in general unbounded) normal operators defined in a Hilbert space & (cf. Nagy [9] p. 50). DEFINITION 1. Let N1 and N2 be normal operators and {K1(z)} and {K2(Z)I their corresponding canonical resolutions of the identity. We say that N1 and N2 permute if Kl(zi)K2(z2) = K2(z2)Kl(zi) for each complex z1 and z2 It should be pointed out here that this definition is, by a theorem of B. Fuglede [4], equivalent to the usual definition of permutability in the case where N1 and N2 are bounded (N1N2 = N2N1) or in the case where N1 is bounded and N2 is unbounded (N1N2 _ N2N1). DEFINITION 2. We say that the operators N1, N2 and N have the property P if they are normal operators and if N = N1N2 = N2N1 .

On Systems of Linear Differential Equations

with U a column vector and A and P n-square matrices. The transformation U = TU, by a unimodular matrix T is easily seen to result in an equation in U, of form (1), in which the coefficient of A is T-1AT. It is known [1] that if the elements of A and its characteristic roots are holomorphic in a closed bounded region R, then there exists a matrix T unimodular in R such that T-1AT is in triangular form. It may be remarked that the theorem is also true for the case R is unbounded.2 It is easily verfied that the method of [1] may be extended to this case without significant alteration. The only possible difficulty is the requirement [1; p. 470] of a function whose finite expansion3 is specified at a set of points which in this case may not be finite (although without a finite accumulation point). However, the existence of such a function is established by a theorem of Mittag-Leffler [2; pp. 5-6]. Another proof which may be extended to the unbounded case is that given in [3] for general principal ideal rings. It is known that the ring of all functions holomorphic in an unbounded region satisfies all the postulates of a principal ideal ring except the infinite chain condition [Cf. 4; p. 351]. However, the proof of [3] makes no use of the chain condition and so may be applied immediately. It will accordingly be assumed that the coefficients of the original equation satisfy the above conditions. In the following it will be supposed that the transformation has already been performed, so that in (1) A is triangular.

The integral representation of unbounded self-adjoint transformations in Hilbert space

That is, we establish the integral representation of A by means of the resolution of the identity Ex(A) corresponding to A (for definitions, see below). The facts in question for the case of bounded transformations have been known in substance since the appearance in 1906 of Hilbert's memoir on integral equations. The unbounded case on the other hand has received attention only in recent years.t Each of the three methods given to establish (1) for unbounded transformations is based to a certain extent on the theory of bounded transformations, indeed to the extent of making use of the transformation (A -iE)-'; but none of these methods exploits formula (1) for the bounded case and its immediate consequences systematically. Our purpose is to show that by doing so the proofs can be shortened considerably. The fundamental idea of the first proof is to establish the existence of an orthogonal system of closed linear manifolds sweeping out the whole space and such that A behaves like a bounded transformation in each manifold considered as a Hilbert space. Thus for each manifold, formula (1) is valid; the behavior of A in the original space is obtained from its behavior in each manifold by a simple synthesis. The motivating idea of the second proof is to introduce a bounded selfadjoint transformation D whose resolution of the identity is topologically

A Property of Unbounded Continua, with Applications

1. A great part of the theory of continua is valid only if the continua under discussion are bounded. This is primarily due, of course, to the fact -that for unbounded sets it is not necessarily true that every infinite set has at least one limiting point. Hence such fundamental theorems as the Cantor divisor theorem and the existence of various kinds of irreducible continua fail.t However, many theorems which are true only for bounded continua are readily generalized for the unbounded case by the imposition of suitable restrictions. The favorite tool used for such extensions has been the method of inversion.t It is the purpose of this article to call attention to a property of unbounded continua (:? 2) which is useful in investigations of this kind and to give applications of it to irreducible continua and the separation of the plane. It is possible that this property has been noted before by other students of the subject (Cf. the article by Knaster and Kuratowski referred to above and the article by Szymanski referred to in ? 5), but, to judge by the literature, it has been discarded as of no importance. The material of ?? 2-5 is valid in any metric space for which the Bolzano-Weierstrass property that every bounded infinite set has at least one limiting point holds; for the sake of brevity such a space will be called a W-space.