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Uniform Boundedness Principle for Unbounded Operators

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Uniform Boundedness Principle for Unbounded Operators UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. Uniform boundedness principle; closed graph theorem. Quit and for all scalars a. A subset M of a vector space X is said to be α-convex (0 < α ≤ 1) if a1/αx + (1 − a)1/αy 2 M for all x; y 2 M and for all scalars 0 ≤ a ≤ 1 (see: [2]). Every topological vector space is Hausdorff. An F -space is a complete metrizable topological vector space. A Frechet space is a locally convex F -space (see: [3]). The notations int A and cl A will denote interior and closure of a set A respectively. 2. Uniform Boundedness Principle Theorem 2.1 (Uniform Boundednss Principle). Let X be an F -space, Y be a vector space and p be an α-seminorm on Y (0 < α ≤ 1). Let (Ti)i2I be a family of linear mappings from X into Y . Suppose fp(Tix): i 2 Ig is bounded for each x 2 X. Then there is an α-seminorm balanced open neighbourhood U of 0 in X, and there is a dense subset A of U such that fp(Tix): i 2 I; x 2 Ag is bounded and fax : x 2 A; a is a scalarg is a dense linear subspace of X. Proof. To each positive integer k, define a set Ak = fx 2 X : p(Tix) ≤ k; for all i 2 Ig. As p is an α-seminorm, Ak is α-convex and balanced; so, cl A is α-convex and balanced. This implies that int cl Ak is also α-convex and symmetric. Indeed, int cl Ak = int cl(−Ak) = − int cl Ak and if 1/α 1/α 0 < a < 1, then a int cl Ak+(1−a) int cl Ak is open in X and it is contained in cl Ak, and hence 1/α 1/α JJ J I II it is contained in int cl Ak. If x 2 int cl Ak 6= ;, then 0 = (1=2) x + (1 − 1=2) (−x) 2 int cl Ak, and so int cl Ak is balanced [3, Theorem 1.13(e)]. Thus int cl Ak is balanced if it is nonempty. Go back 1 Since X = [j=1 Ak, int cl Ak 6= ; for some k. Take U = int cl Ak and Z = fax : x 2 Ak \ U; a is a scalarg. If x; y 2 Z, there is a scalar a > 0 such that ax; ay 2 Ak \ U; this implies that Full Screen 1/α 1/α (1=2) ax+(1−1=2) ay 2 Ak \U, and hence x+y 2 Z. This shows that Z is a linear subspace of X. If a nonzero x 2 X and a neighbourhood V of 0 such that V ⊂ U are given, then let us find Close a scalar a > 0 such that ax 2 U. We can find a balanced neighbourhood W ⊂ V of 0 such that ax+aW = a(x+W ) ⊂ U ⊂ cl Ak. So, there is an element y 2 W ⊂ V such that ax+ay 2 Ak \U. Quit Thus x + y 2 Z \ (x + V ). This proves that Z is a dense linear subspace of X. Since U ⊂ cl Ak and U is open, U \ Ak is dense in U. So, we take A = U \ Ak. This completes the proof. The arguments imply the following results. Theorem 2.2. Let X be an F -space, fYigi2I be a collection of vector spaces and pi be an α- seminorm on Yi for a fixed α, (0 < α ≤ 1). Let fTigi2I be a family of linear mappings Ti : X ! Yi. Suppose fpi(Tix): i 2 Ig is bounded for each x 2 X. Then there is an α-convex balanced open neighbourhood U of 0 in X, and there is a dense subset A of U such that fpi(Tix): i 2 I; x 2 Ag is bounded and fax : x 2 A; a is a scalarg is a dense linear subspace of X. Theorem 2.3. Let X be an F -space, and Y be a topological vector space. Suppose fpjgj2J is a family that defines the topology of Y , where each pj is an αj-seminorm on Y , (0 < αj ≤ 1). Let fTigi2I be a family of linear mappings from X into Y such that fTix : i 2 Ig is bounded for each x 2 X. Then for each j 2 J, there is an αj-convex open neighbourhood Uj of 0 in X, and there is a dense subset Aj of Uj such that fpj(Tix): i 2 I; x 2 Ajg is bounded. Corollary 2.4. Let X be a Banach space, Y be a normed space, and fTigi2I be a family of linear mappings of X into Y . Suppose fTix : i 2 Ig is bounded for each x 2 X. Then there exists a dense subset A of the unit open ball in X such that fTix : i 2 I; x 2 Ag is bounded. JJ J I II The following theorem is an application of our principle. Go back ∗ Theorem 2.5. Suppose fTigi2I and fTi gi2I are two families of linear mappings from X into Full Screen Y and from Y 0 into X0, respectively, when X and Y are normed spaces, and when X0 and Y 0 ∗ 0 are respective duals, and suppose they satisfy (f ◦ Ti)x = (Ti f)(x) for all f 2 Y , x 2 X, i 2 I. Close ∗ 0 ∗ Suppose fTi f : i 2 Ig is bounded for each f 2 Y . Then fTigi2I and fTi gi2I are equicontinuous on X and Y 0, respectively. Quit Proof. By our principle, there is a convex open neighbourhood U of 0 in Y 0 and there is a dense ∗ subset A of U such that fTi f : i 2 I; f 2 Ag is bounded. We lose no generality by assuming that 0 ∗ U is the open unit ball in Y . There is a k > 0 such that kTi fk ≤ k for all i 2 I; for all f 2 A. Then ∗ ∗ j(f ◦ Ti)xj = j(Ti f)(x)j ≤ kTi fkkxk ≤ kkxk for every f 2 A; i 2 I; x 2 X That is sup jf(Tix)j ≤ kkxk for every x 2 X; i 2 I: f2A This implies that sup jf(Tix)j ≤ kkxk for every x 2 X; i 2 I: f2U Therefore kTixk = sup jf(Tix)j ≤ kkxk for every x 2 X; i 2 I kfk≤1 ∗ ∗ Thus (Ti)i2I is equicontinuous on X. Note that kTik = kTi k for all i 2 I (see [1]). So, (Ti )i2I is also equicontinuous. Remark 2.6. If I is finite, then fT ∗(f): i 2 Ig is bounded for every f 2 Y 0. JJ J I II i The argument used in the proof of Theorem 2.5 implies the next theorem Go back ∗ Theorem 2.7 (A General form of Hellinger-Toeplitz theorem). Suppose fTigi2I and fTi gi2I Full Screen are two families of linear mappings from a Hilbert space H1 into a Hilbert space H2 and they satisfy ∗ ∗ hTix; yi = hx; Ti yi for every i 2 I, x 2 H1, y 2 H2. Suppose fTi y : i 2 Ig is bounded for every Close y 2 H2 or fTix : i 2 Ig is bounded for every x 2 H1. Then fTigi2I is equicontinuous on H1 and ∗ fTi gi2I is equicontinuous on H2. Quit 3. A Closed Graph Theorem Theorem 3.1 (Closed Graph Theorem). Let fTigi2I be a family of linear mappings from an F -space X into an F -space Y . Suppose the topology of Y is defined by a countable collection 1 fpngn=1, where each pn is an αn-seminorm, (0 < αn ≤ 1). Suppose each Ti has closed graph in X × Y . Assume further that fTix : i 2 Ig is bounded for each x 2 X. Then fTigi2I is equicontinuous on X. Proof. By a classical closed graph theorem, each Ti is continuous. By Theorem 2.3, for each given n = 1; 2;::: , there exists an open neighbourhood Un of 0 in X and there is a dense subset An of Un such that pn(Tix) ≤ kn for all i 2 I, for some kn > 0; for all i 2 I and for all x 2 An.
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