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LM6144AIM/NOPB 产品描述及参数 - TI（德州仪器）
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TI（德州仪器）
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运算放大器
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SOIC-14
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TEXAS INSTRUMENTS LM6144AIM/NOPB. 芯片, 运算放大器, 17MHZ, 25V/ US, SOIC-14
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LM6144AIM/NOPB数据手册
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Math. Ann.242, 85-06 (1979)
k lm
Anm
@ by Sprmger-Verlag 1979
A Geometric Proof of the Spectral Theorem
for Unbounded Self-Adjoint Operators
Herbert Leinfelder
Universit~it Bayreuth, Fachbereich Mathernatik und Physik, Postfach 3008, D-8580 Bayreuth,
Federal Republic of Germany
tn this note we are concerned with unbounded self-adjoint operators in a Hilbert
space. Denoting such an Operator by A we give a direct and geometric
demonstration of the facts associated with the formula
A= ~ 2dE(2).
-oo
That is, we establish directly the well-known spectral theorem for unbounded
self-adjoint operators using only simple geometric intrinsic properties of Hilbert
space.
Of course the fundamental facts about the spectral representation of bounded
as well as unbounded operators have been known in substance since the
appearance in 1906 [3] of Hilbert's memoir on integral equations and in 1929 [8]
of von Neumann's fundamental paper on unbounded operators. Since that time
many papers have been published in this subject using a variety of methods. Some
of these methods apply only to bounded operators, while others are suited to the
general (unbounded) case. However, all but a few of these methods use techniques
and principles which lie outside of Hilbert space theory proper, such as Helly's
selection principle, Riesz's second representation theorem and so on. For further
examples of technique and for a comprehensive list of references on the spectral
theorem we refer the reader to [1], p. 927 and [5].
It was not until 1935 that Lengyel and Stone [5] gave a new proof of the
spectral theorem which was strictly elementary in the sense that it depended only
upon intrinsic properties of Hilbert space. Their paper dealt with the case of
bounded operators and the authors remarked that they could not handle the
general case in the same way. In their introduction they wrote: "Indeed our
method, which requires the study of powers of an operator, is not suited to the case
of unbounded operators" ([5], p. 853). Just this sentence stimulated the author to
try to handle the general case in the same way and indeed it is possible.
The fundamental idea of our proof of the spectral theorem as well as that of
Lengyel and Stone consists in considering invariant subspaces
F(A,2)=
{xenlxeD(A"),
]lA"xl[ <2". Itxll for n = 1,2, 3 .... }.
0025-5831/79/0242/0085/$02.40
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