## Locating the range of an operator on a Hilbert space (1992)

Venue: | Bull. London Math. Soc |

Citations: | 3 - 3 self |

### BibTeX

@ARTICLE{Bridges92locatingthe,

author = {Douglas Bridges and Hajime Ishihara},

title = {Locating the range of an operator on a Hilbert space},

journal = {Bull. London Math. Soc},

year = {1992},

volume = {24},

pages = {599--605}

}

### OpenURL

### Abstract

In classical operator theory it is taken for granted that we can project onto the closure of the range of an operator T on a Hilbert space H. In a constructive development of operator theory, to which this note is a contribution, this projection exists if and only if ran(r), the range of T, is located, in the sense that the distance

### Citations

129 |
and Complex Analysis McGraw-Hill
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- 1987
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Citation Context ...t (xn) be a sequence of unit vectors in X such that \\Tnxn\\-+co as «->oo. Then there exists xeX such that the sequence {\\Tnx\\) is unbounded. Proof. The proof is similar to a standard classical one =-=[11]-=-. THEOREM 4 (Hellinger-Toeplitz Theorem). A selfadjoint linear operator on a Hilbert space is sequentially continuous. Proof. Let T be a selfadjoint operator on a Hilbert space H, (xn) a sequence in H... |

60 |
Foundations of Constructive Mathematics
- Bishop
- 1967
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Citation Context ... exists a value N such that aN = 1. To see this, consider the following Brouwerian counterexample. (For a discussion of the nature and role of Brouwerian counterexamples in mathematics, see page 3 of =-=[6]-=-.) EXAMPLE 1. Let a be a real number such that -*(a = 0), and define a selfadjoint operator T on the Hilbert space U by Tx = ax. (We must distinguish carefully between the expressions '-i(a = 0)' and ... |

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A constructive look at positive linear functionals on
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- 1981
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Citation Context ... characteristic function of {0} is integrable with respect to the functional calculus measure for \T\ (compare pages 315-316 of [12]). In general, the following lemma, which appears as Theorem 1.1 of =-=[2]-=-, provides an approximate polar decomposition which is a satisfactory constructive substitute for the exact one. LEMMA 1. Let T be an element of s#{H), and e > 0. Then there exists a partial isometry ... |

7 |
Constructive compact operators on a Hilbert space
- Ishihara
- 1991
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Citation Context ...(3 n *), the kernel of T*. (For an example to show that the existence of the adjoint of a bounded operator on a Hilbert space is not automatic in constructive mathematics, see Brouwerian Example 3 in =-=[7]-=-; see also Example 2 below.) This much is neither surprising nor particularly interesting. Of more interest is the observation that, constructively, the existence of the projection on ker(T*) is no gu... |

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A constructive treatment of open and unopen mapping theorems
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- 1989
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Citation Context ...nciple, such as the limited principle of omniscience [1, 6]. Then, in view of the last corollary, we have a Brouwerian counterexample to the classical open mapping theorem. Despite extensive analysis =-=[5]-=-, no such counterexample has been found. So the question with which we began this section remains unanswered and intimately associated with the constructive status of the open mapping theorem in its f... |

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Linear mappings are fairly well-behaved
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Citation Context ... (iv) => (ii). COROLLARY. If T is an element of $0{H) with complete range and located kernel, then ran(J') is located. Proof This is an immediate consequence of Theorem 1 and the fact, established in =-=[4]-=-, that any linear mapping from a normed space onto a Banach space is wellbehaved. Example 1 shows that we cannot remove from Theorem 1 the condition that T is well-behaved; and that, in the above coro... |

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Operator ranges, integrable sets, and the functional calculus
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- 1985
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Citation Context ...is section we show that the answer to this question is closely connected with the constructive status of the open mapping theorem. We begin with two elementary lemmas, the first of which is proved in =-=[3]-=-. LEMMA 5. If M and N are orthogonal linear subsets ofH such that M+N is dense in H, then M and N are both located. A linear mapping T of a normed space X into a normed space is said to be open if the... |

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A constructive closed graph theorem
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Citation Context ... principle: to see this, consider the linear mapping of Example 1. Pursuing more directly our discussion of the question heading this section, we now state a remarkable lemma, whose proof is found in =-=[8]-=-. LEMMA 6. Let T be a linear mapping of a Banach space X into a normed space Y, and let (xn) be a sequence in X converging to 0. Then for all positive numbers a,/? with a < /?, either || Txn\\ > a. fo... |

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Meaning and information in constructive mathematics
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Citation Context ... the above corollary, we cannot omit the hypothesis that T has complete range. (Note that, for any real number a, the subset Ua = {ax:xe 05} of U is complete if and only if either a = 0 or a ^ 0; see =-=[10]-=-.) The following example shows that in Theorem 1 and its corollary we need T to have an adjoint defined on the whole space H. EXAMPLE 2. Let a be a real number such that ->(a = 0), and let H be the Hi... |

1 |
Constructive analysis, Grundlehren
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- 1985
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Citation Context ...th constructive versions of the open mapping theorem. We shall also discuss how far our main theorem on locating ranges is the best possible within BISH. We shall assume that the reader has access to =-=[1]-=-, Chapters 4 and 7 of which provide the basic material on metric, normed and Hilbert spaces upon which our work is founded. We denote by <x, y} the scalar product of two vectors x and y in a Hilbert s... |

1 |
Constructive existence of Minkowski functionals
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Citation Context ...onclude that ran(!T*) 4- ker(r) is dense in H. Since ker(!T) is the orthogonal complement of ran(!T*), it follows from Lemma 5 that ran(7) is located. Theorem 2 should be compared with Corollary 1 of =-=[9]-=- .IfTisa sequentially open linear mapping of a normed space onto a Banach space, such that T(B(0,1)) is located, then ker(r) is located. It follows from Theorems 1 and 2 that a sequentially open eleme... |