Norms Of Projections Research Articles

Norms of some projections on C[a, b

Norms of some projections on C[a, b

Projections from 𝐿^{𝑝}(𝐺) onto central functions

Let G G denote a locally compact [ F C ] B − [FC]_B^ - -group. Then for every 1 ⩽ p ⩽ ∞ 1 \leqslant p \leqslant \infty there is a projection from L p ( G ) {L^p}(G) onto the B B -central functions in L p ( G ) {L^p}(G) .

Minimal projections on hyperplanes in sequence spaces

If Iz is a closed linear subspace in a Banach space X, then a projection of X onto Y is a bounded linear map P: X -> Y such that Py = y for all y E Y. If such a projection exists, then Y is said to be complemented in X. In this case, there is some interest in discovering whether there exist projections of minimal norm, and if so, what their properties are. Many applications of projections occur in numerical anMysis and approximation theory, for/~x can be regarded as an approximation to x in IT. The quality of this approximation relative to the best approximation is governed by the inequality

A characterization of Hilbert space

A real Banach space E of dimension _3 is an inner product space iff there exists a bounded smooth convex subset of E which is the range of a nonexpansive retraction. De Figueiredo and Karlovitz [3] have shown that if E is a strictly convex real finite-dimensional Banach space and dim E> 3 then there can exist no bounded smooth nonexpansive retract of E unless E is a Hilbert space. (A subset F of E is a nonexpansive retract of E if it is the range of a nonexpansive retraction r: E-F.) This is a consequence of their more general result that if E is reflexive and a convex nonexpansive retract of E has at a boundary point xo a unique supporting hyperplane xo+H then H is the range of a projection of norm 1. As they have pointed out, the latter theorem fails in nonreflexive spaces (the unit ball of C[O, 1] furnishes a counterexample). Nevertheless, their first result is true in general: THEOREM. Suppose E is a real Banach space with dim E> 3. Then E is an inner product space iff there exists a bounded smooth nonexpansive retract of E with nonempty interior. We separate out of the proof of the theorem a lemma, valid in all real Banach spaces: LEMMA. Suppose F is a bounded smooth closed convex subset of a real Banach space E and F has nonempty interior. Then given disjoint bounded closed convex sets M and K in E with K compact, there exist p E E and 2>0 such that Kcp+)LF and (p+ 2F) rnM= 0. PROOF OF LEMMA. Clearly the hypotheses and conclusions of the lemma are invariant if K and M are translated by the same vector; thus without loss of generality we may assume 0 E K. Similarly, we may also assume 0 E int F. Since K is compact and M is closed, a basic separation theorem for convex sets assures the existence of a closed hyperplane H which strictly separates M and K; that is, there exist we E*, c e R' Received by the editors June 26, 1972 and, in revised form, August 21, 1973. AMS (MOS) subject classifications (1970). Primary 46C05.

On some types of maximal abelian subalgebras

Let H be a Hilbert space and B( H) the algebra of all bounded linear operators on H. It is known that there are two kinds of maximal abelian sub-algebras in B( H), to one of which there exists a unique faithful normal projection of norm one from B( H) and to the other any projection of norm one is singular. Any maximal abelian subalgebra A contains a projection e such that Ae is a maximal abelian subalgebra of B( eH) of the first kind and A(1 − e) is the one of the second kind in B((1 − e) H). This will be generalized to an arbitrary von Neumann algebra together with the existence problem of those kinds of maximal abelian subalgebras.

Conditional expectations in von Neumann algebras

Let M be a von Neumann algebra and N its von Neumann subalgebra. Let ϑ be a faithful, semifinite, normal weight on M + such that the restriction ϑ ¦ N of ϑ onto N is semifinite. The first main result is that N is invariant under the modular automorphism group σ t ϑ associated with ϑ if and only if there exists a σ-weakly continuous faithful projection ϵ of norm one from M onto N such that ϑ ̇ (x) = ϑ ̇ ∘ ϵ(x) for every xϵ M ϑ. The second result is that a von Neumann algebra M is finite if and only if any maximal abelian self-adjoint subalgebra of M is the range of a σ-weakly continuous projection of norm one. This result is an answer for the question which Kadison raised in the author's talk at the International Congress of Mathematicians in Nice, 1970.

Contractive projections in continuous function spaces

Let C(K) be the Banach space of real-valued continuous functions on a compact Hausdorff space with the supremum norm and let X be a closed subspace of C(K) which separates points of K. Necessary and sufficient conditions are given for X to be the range of a projection of norm one in C(K). It is shown that the form of a projection of norm one is determined by a real-valued continuous function which is defined on a subset of K and which satisfies conditions imposed by X. When there is a projection of norm one onto X, it is shown that there is a one-to-one correspondence between the continuous functions which satisfy the conditions imposed by X and the projections of norm one onto X. Using well-known results of Nachbin, Goodner and Kelley (see [2, p. 2]), it is easily demonstrated that if S is an extremely disconnected compact Hausdorff space, then X, a closed subspace of C(S), is the range of a norm one projection in C(S) iff X itself is isometric to the continuous functions on an extremely disconnected compact Hausdorff space. The main purpose of this research is to give necessary and sufficient conditions for a closed separating subspace X of C(K), K a compact Hausdorff space, to be the range of a norm one projection in C(K). Theorem 3.1 supplies these conditions. Proposition 1.2 shows that the form of each contractive projection onto X can be given in terms of a real-valued continuous function, determined by the projection, which is defined on a subset of K. This continuous function satisfies certain conditions which depend only on the subspace X. Theorem 3.1 also shows that when there is a contractive projection onto X there is a one-to-one correspondence between contractive projections onto X and the continuous functions which satisfy these conditions imposed by X. We give another proof to the fact, proved in [3], that the Banach space Y is isometric to the range of a contractive Received by the editors June 11, 1971 and, in revised form, February 9, 1972. AMS 1970 subject class{flcations. Primary 46E15, 47B99.

On a Lemma of Milutin Concerning Averaging Operators in Continuous Function Spaces

We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $S’$, having the same topological weight as S, by a map $\varphi$ which admits a regular linear operator of averaging, i.e., a projection of norm one of $C(S’)$ onto ${\varphi ^ \circ }C(S)$, where ${\varphi ^\circ }:C(S) \to C(S’)$ is the isometric embedding which takes $f \in C(S)$ into $f \circ \varphi$. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $S’$ can be taken to be the Cantor space ${\{ 0,1\} ^\mathfrak {m}}$, where $\mathfrak {m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $\varphi :S \to T$, the operator $u:C(S) \to C(T)$ is a regular averaging operator for $\varphi$ if and only if u has a representation $uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $\theta :T \times [0,1] \to S$.

On a lemma of Milutin concerning averaging operators in continuous function spaces

We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $S’$, having the same topological weight as S, by a map $\varphi$ which admits a regular linear operator of averaging, i.e., a projection of norm one of $C(S’)$ onto ${\varphi ^ \circ }C(S)$, where ${\varphi ^\circ }:C(S) \to C(S’)$ is the isometric embedding which takes $f \in C(S)$ into $f \circ \varphi$. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $S’$ can be taken to be the Cantor space ${\{ 0,1\} ^\mathfrak {m}}$, where $\mathfrak {m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $\varphi :S \to T$, the operator $u:C(S) \to C(T)$ is a regular averaging operator for $\varphi$ if and only if u has a representation $uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $\theta :T \times [0,1] \to S$.

Feldspar-liquid equilibria in peralkaline liquids; the orthoclase effect

Recent published discussions of pantellerites and related experimental liquids have described compositional differentiation trends in terms of normative Q, or, and ab. This norm projection has at least three intrinsic deficiencies when used to depict paralkaline liquid compositions: (1) It exaggerates potential free silica in the liquid if ns forms part of the total composition. (2) K 2 O is arbitrarily allotted to or; peralkalinity is thus expressed only in terms of Na 2 O, which is thereby lost in the projection. (3) By ignoring part of the Na 2 O in peralkaline liquids the projection conceals an important relationship in the alkali balance between the liquids and their feldspar phenocrysts--the orthoclase effect.

Some uncomplemented function algebras

1. Let X be the unit circle Iz I = 1, and A the disc algebra of functions on X having continuous extensions to I z ? < 1 analytic for I z I < 1. Then it is known [12] that there is no bounded projection of C(X) onto A; alternatively, A is uncomplemented in C(X). To what extent is this a general occurrence? Specifically, if A is a closed nonselfadjoint(2) subalgebra of C(X), X compact, is A uncomplemented in C(X)? Only some partial results will emerge here. From recent results of Bishop [1], extended to the nonmetric case by Bishop and deLeeuw in [2], we obtain the curious fact that if X is the Silov boundary of A, and T is any bounded operator on C(X) acting as the identity on A, then T I || = 11 T |1, where I is the identity operator; alternatively, any operator S (= I T) annihilating A has || I + S || = 1 + || S ||. As a consequence of this fact one can apply the technique of [12] to show that if X is a compact group and our nonselfadjoint subalgebra A of C(X) is translation invariant, A is uncomplemented; in fact any closed subspace lying between two invariant algebras A1 c A2, with the set of conjugates A1 t A2, is uncomplemented (?3). And this applies equally well to invariant subalgebras A1, A2 of(3) CO(X), where X is a locally compact abelian group (?4); but both proofs are technically complicated, and shed no light on the situation in general. In what follows we shall consider a slightly more general setting in which A is a subalgebra of CO(X), X locally compact; since we shall be concerned with estimating the norms of projections, the usual adj unction of an identity does not lead easily to a reduction to the compact case. The author is indebted to K. deLeeuw for several helpful comments and, in particular, Theorem 4.1 is due to deLeeuw.

On gradient mappings in Banach spaces

Thus x->D(x, *) is a mapping of V into the space E* conjugate to E, and is called a gradient mapping. It is said to be compact, if the image of each bounded set of V has a compact closure in E*. Recently E. H. Rothe [1 ] gave a (necessary and) sufficient condition for the compactness of gradient mappings under the condition that E has property (P), and showed that every reflexive Banach space with a basis has it. Here E is said to have property (P), if there exists a sequence {{i} of linearly independent elements of E* and a positive number M with the following properties: the closed linear span of { i} coincides with E* and for each positive integer n there exists a linear projection of norm at most M on the intersection nli Ni where Ni= {xCE I i(x) = 0 1. In this note a theorem with the conclusion that the gradient mapping is compact is proved under conditions differing somewhat from those of Rothe, and without using anything like the property (P).

Projective Test Norms and Their Use in Case Studies

The problem of evaluating projective test data for an individual patient in subjective and/or in statistical terms continues to be much discussed. Some clinicians feel that use of statistical norms obscures much of value about the individual protocol. Also the test materials demand that the individual project himself always as a changing being. In ;he case of the Family Attitudes Test (Jackson, 1952), the small child is required to identify with a child of its own age pictured in a rigid form which repeats the parent-sibling situation. For older patients the Four Picture Test (van Lennep, 1958) permits freedom of choice of the sequence in which the cards are responded to but the main figure is male, boy or a man. This test requires that a girl identify with these male figures. To facilitate projection we used a procedure which simulates the comic strip. S is asked to arrange in a sequence all or part of eight plates from Sy~nonds' test (1948) on which a boy is shown. Then S tells a story which is recorded. This procedure has been used for several years for each case study. The present research was based on 150 protocols of male youngsters, ranging from 8 to 18 yr. To establish norms the following response characteristics were studied: (a) number of picrures used for the story by a child at successive ages and for children in each age-group; (b) which pictures were chosen and which ones excluded by children of different age-groups; (c) position of each picture within stories; (d) initial and final pictures; (e) titles for stories; and (f) variation of the significance given to each single picture (i.e., father and mother image, isolation amidst sky-scrapers, etc.). The results of these analyses were evaluated for individual cases. Common solutions could be distinguished from rare solutions within a given age-group. Thus the story as a whole not only revealed the child's problems but also his personal dynamics, his aspiration level, his fears, his relation to the world around him, and so on. The same form of the test given a particular individual at different ages then, tells about his changing attitudes and concepts. A parallel study is planned with stories made up by girls, using nine pictures of a girl figure. REFERENCES

On the projection of norm one in $W^*$-algebras, III

On the projection of norm one in $W^*$-algebras, III

On the product projection of norm one in the direct product of operator algebras

On the product projection of norm one in the direct product of operator algebras

On the projection of norm one in $W^{\ast}$-algebras, II

On the projection of norm one in $W^{\ast}$-algebras, II

On the projection of norm one in $W^ *$-algebras

On the projection of norm one in $W^ *$-algebras

A theorem of the Hahn-Banach type for linear transformations

Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with the question of the existence of projections of norm one, and moreover that the nature of the space where the transformations take their values is much more important than that of the space where the transformations have to be defined. It is not known, as far as we can say, what are the precise conditions for the possibility of extending a transformation without disturbing its linearity, continuity, and norm. In this paper we shall give a necessary and sufficient condition, which refers only to the space where the transformation takes its values and does not involve the transformation itself or the space where it is to be defined, for such an extension to be possible: the condition is expressed in terms of a certain "binary intersection property" of the collection of spheres of the normed space (see Theorem 1). After that we proceed to the study of the structure of real normed spaces whose collections of spheres have this property. A first step in this direction is given by the theorem asserting that these normed spaces (provided they contain at least one extreme point in the unity sphere) are simply those that can be made into complete vector lattices with order unity in such a manner that the norm derived from the order relation and the order unity in the natural way is identical to the given norm (see Theorem 2, and Theorem 3 for the finite-dimensional case). In this connection we point out a conjecture which we have not been able to settle, namely that, if the collection of spheres of a normed space has the binary intersection property, then its unity sphere must contain an extreme point. By using some results of S. Kakutani and M. H. Stone, we establish the connection between the normed spaces having the binary intersection property and the spaces of real continuous functions over certain compact Hausdorff spaces (see Theorem 4), or the complete Boolean algebras (see Theorem 6).

Convex Regions and Projections in Minkowski Spaces

By a well known theorem of Hahn-Banach every linear functional of norm M defined over a closed linear subspace of a Banach space' can be extended linearly to the entire space without increasing its norm. For operations the corresponding problem takes the following form: Given a linear operation u(x) whose domain of definition is a closed linear subspace of a Banach space B, and whose range lies in a Banach space B2, does there exist an operation U(x) defined over B1, with range in B2 and which coincide with u(x) over the subspace? How small can the norm of the extended operation be made? In the particular case where u(x) = x, B1 = domain of definition of u; the existence of an extension U(x) is equivalent to the existence of a complementary subspace or, as F. J. Murray2 has shown, to the existence of a projection of B, on the subspace. Conversely, if A(X) is such a projection and u(x) any linear operation, the linear operation U(X) = u(A(X)) is an extension of u(x) and its norm is < 11 u 11.11 A 11. The question of extension is thus equivalent to the discussion of best projections, i.e. of projections of least norm. It is relatively easy to exhibit examples of Banach spaces for which certain closed subspaces have no projections; even more, F. J. Murray3 has shown that this occurs within the function spaces L, and the sequence spaces lp! For such subspaces an extension of an operation is not always possible. If the Banach space is finite dimensional, i.e. if we are dealing with a Minkowski space, of n dimensions say, the existence of projections is trivial but the discussion of the best possible projections (i.e. those with a minimal norm) is interesting and may help to explain why, in certain infinitely dimensional spaces, projections fail to exist. We give in this paper a complete answer to the question of best possible projections in the case where the dimensionality of the subspace on which we project is by 1 less than the dimensionality of the entire space. The results obtained lead to some interesting theorems on convex regions; they are considered in section 6.