TOKYO MATHEMATICAL BOOK SERIES
VOLUME
IV
SPECTRAL THEORY IN
THE
HILBERT
SPACE
BY HIDEGORO NAKANO
1953 JAPAN SOCI...
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TOKYO MATHEMATICAL BOOK SERIES
VOLUME
IV
SPECTRAL THEORY IN
THE
HILBERT
SPACE
BY HIDEGORO NAKANO
1953 JAPAN SOCIETY FOR THE PROMOTION OP SCIENCE Ueno Park, Taitoku, Tokyo, Japan
PREFACE
Spectral theory was considered first
symmetric operators spectral theory bounded
on a sequence space.
take a new form*
symmetric
operators,
by D. Hllbert
about bounded
J. von Neumann made this
He generated this theory
defining abstractly
to un-
the Hilbert space.
In this new style spectral theory was discussed by many mathematicians: P. Riesz, M. H. Stone, etc.
On the other hand
as an a Detract ion
near spaces* space.
in semi-ordered li-
a spectral theory was born
of the spectral theory
in the Hilbert
This spectral theory
is constructed
by means of generated,
considering
every spectrum
as
Stieltje*s integral,
On the contrary by the author,
there is
a continuous one.
another type of spectral theory
which is constructed
established
by means of generated Riemann in-
tegral in topological spaces considering every spectrum as a point one*
This new spectral theory is explained
precisely
in
Volume II of this
Series, Modern Spectral Theory, Tokyo (1950), in the name of the second
spectral theory. dpeJttraltheorie,
The auther has remarked in an earlier paper, Bine Proc. Phys.-Math. Soc. Japan, 2)(l94l) 485-511,
that
this spectral theory also is applicable to the Hilbert space, but it is
not well known.
The purpose of this book is to make this new spectral
theory revive in the Hilbert space.
In this text spectral theory is constructed firstly about elements of the Hilbert space, and as its application, we consider the theory of the normal operators.
Furthermore,
*{ctral
by means of this new
ii
spectral theory, we can discuss
not only
the mean ergodic theorem ob-
but the individual ergodic theorem
tained by J. von Neumann,
by J. D. Birkhoff , as a problem in the Hilbert space.
obtained
Readers will
be more interested, if they read this text, comparing Volume II of this
Series,
This book was written
matical Book Series. to publish this book.
during 1950-51 as Volume IV of Tokyo MatheBut financial difficulties made it impossible
Recently
I
have obtained financial help from
Education Ministry of Japan, and it has become possible to publish this book.
I
express my warmest thanks to those who have made effort to
publish this book.
Sapporo,
June 24,
1953
iii
TABLE OF CONTENTS
INTRODUCTION
1
CHAPTER 1.
2
.
3.
4. 56.
7. 8.
9. 1
.
Completion Orthononnal Systems Dimensions of Hilbert Spaces Weak Convergence
11 .
13 . 14.
15 16. 17. 18. 19 . 20. 21.
.
'7
.
J28. 29* 30.
31.
$33. $34*
J35
536.
.20
*
.25 .29
23
*
32
SPECTRAL THEORY
II.
41 44 .49
53 .60
Proper Spaces Integration Integration by Elements.... Relative Spectra Properties of Relative Spectra Properties of Rings. . . . Ring Convergence.
66 72 79 85 .89 ,
94
DILATATOR ANALYSIS
Dilatators
101 108 116
Speotralitiea Spectralization of Dilatators Calculus of Dilatators
123 134 142 147 152 165
D-convergence Measurable Functions General Spectralities Integral Operators Functions of Dilatators
NORMAL OPU31ATORS
IV.
Fundamental Definitions* , Regular Part Properties of Symmetric Operators Specialization of Normal Operators Graphs of Operators Regularity. * Self -ad joint Operators* ..,.* Isometric Operators .**..... ,
|36. 837.
*
, , .
Projection Operators Calculus of Projection Operators Systems of Projection Operators Rings of Projection Operators..,..
CHAPTER
532.
...,.14 1?
35
CHAPTER III. |22 23. 24 . 25. 6.
8 .10
Operators
CHAPTER
12.
HILBKRT SPACES
I.
, . Linear Spaces * . . . Homed Spaces Adjoint Spaces Inner Product Spaces . . . * Adjoint Spaces of Inner Product Spaces*
,
175 181 ,184
189 195 197 .201
206
TABLE OP CONTENTS
iv
Canonical Form Hermitean Operators
5)9-
^40.
210 214
CHAFm 41 42.
$4546.
49.
CHAPTER
53.
55*
22? 253 23/ 242 251 255 257 266
VI.
ERGODIC THEOREMS
firgodic Theorem of founded Linear Operators
50.
54-
220
Isomorphisms of Specialities Unitary Trans format ion of Dilatators ( L i)-spacea Regular 6 -ideals
47. 48.
52 .
UNITARY INVARIANTS
Multiplicity Resolutions Multiplicity Functions rf-ideals of Sets
J4544.
51.
V.
Isomorphisms of Rings Rings of Uniform Multiplicity
t
Ergodic Theorem of Dilatators Translators Ideal Spaces Reduction Theory Measure Preserving Transformations
PROOK OF
APPENDIX.
RdKEREtCES BIBLIOGRAPh
CC
=
,
276 278 279 285 292 294
298
298 't
299
INTRODUCTION We can modify objects for consideration, and we obtain a notion of
Let ft be a space.
spaces .
be a point or an element of
a
mean that
A
$
,
and
Let
ft
R
*
every element
Oi
( X
and
an element 0i(x
)
X
X by
A
of>
* ? X or X
,
X
implies X
8
subset of X
C
ft
,
A
6
and
,
if 8
^
R
,
A
of
x
R
X
R
about
(*)
x
%
of
if
,
R
into the space
x
:
C
j?
,
* *
into the space of sets of * * A
)
^=
{
,/4
A
by
.
and that
,
R
of
C B or 3 o A
&
,
is called
,
,
X A
to mean that
8
or that
a
is
:
x?
we shall denote by y
COO
^
of
A
\.
as
,4
A
Considering every set of
R
R.
A mapping
;
X
R
/4 X (
\
is called a set system.
/U
AA
for some
X