On the semi-hyponormal n-tuple of operators - CiteSeerX

Integral Equations and Operator Theory Vol.6 (1983)

0378-620X/83/060879-20501.50+0.20/0 © 1 9 8 3 Birkh~user Verlag, Basel

ON THE SEMI-HYPONORMAL n-TUPLE OF OPERATORS

Daoxing Xia

The theory of singular integral model and trace formula is extended to the context of hyponormal or semi-hyponormal ntuple of operators. The spectrum of noncommutative n-tuple of operators is examined.

§i.

INTRODUCTION

The theory of singular integral model of hyponormal operators normal operators well known;

[9], [I0],

[12],

[16],

and trace formula [17], semi-hypo-

[18] or nearly normal operators

[2], [12] is now

and the theme appears with many variations.

We wish to extend this theory to

n-tuple of operators

and to establish a theory corresponding to singular integral of multivariables.

Certainly some mathematicians have extended the

theory in some important cases

(cf.

[3], [4], [8]).

But it seems

that the case which we have examined in this paper Perhaps is a direct one. In §2, we give the definition of semi-hyponormal tuple of operators and its general polar symbols.

Besides,

a special

class of singular integral operator in the space of vector-valued square integrable functions is introduced. In §3, the singular integral model of the semi-hyponormal tuple of operators is established. In §4, the spectrum of semi-hyponormal tors is defined.

tuple of opera-

The relation between the spectrum of semi-hypo-

normal tuple of operators and the joint approximated point spectrum of its general polar symbols is found. In §5, the trace formula of semi-hyponormal tuple is established under certain conditions. theory of principal function

A small part of Pincus'

[i], [2], [3], [4] is generalized to

880

Xia

the s~mi-hyponormal tuple,

tuple

an inequality

case.

similar

Also

for the semi-hyponormal

to Putnam's

inequality

[14] is

proved. In §6, the definitions the hyponormal

DEFINITIONS

In this paper, is the algebra

=(~l,''',~n ) ~,

(~)'

~.

Let

~

INTEGRAL

is a separable

of all linear bounded

is a commuting

be the mapping

QjT = T - ~ j T ~ for all

AND SINGULAR

n-tuple

to

T E ~(~)

QjQk T : QkQjT.

~(~)

complex

Hilbert

operators

in

~(~)

defined

in

space,

~,

operators which

in

commute

by

I

and

If

in

OPERATORS

of unitary

being the set of all operators Qj

corresponding

case are introduced.

§2. ~(~)

and theorems

j = 1,2,..-,n.

A E ~(~),

A m 0

It is evident

that

and

Qj~..Qjm A ~ 0 then A is said to be in for all I g Jl < J2 < • .. < Jm g n, the class SH(~), and the (n+l)-tuple (~,A) is said to be semi-hyponormal. operator linear

~i A

For

combination

ficients

n = i,

of operators

is in

SH(~),

i.e.,

and

B E ~,

then

A E SH(D)

(~i A)

is semi-hyponormal

is semi-hyponormal

[18]. in

For fixed

SH(~)

SH(~)

~,

iff the the

with nonnegative

is a cone in

~(~).

coefIf

B*AB E SH(~). A set of operators basis

of

SH(~) in

BI,''',B N

A = Let

[RI,...,RN]

in

A E SH(~)

there

if for every

SH(~) exist

is called

a

operators

~' such that N ~ B~R.B.. j:l J J J 3~ 3

be the set of all operators

T E ~(~),

for

which J~T = s t - l i m J n~±~ exists. polar

If

T E 3 j+ N 37, J symbols [18] of T

,n ~j-n T~j

(i)

then the operators with respect

to

~j.

J ~$T

are called

It is evident

the

Xia

881

T E 3 j+

that

iff

F~T = st-limN,~ n=l ~j~ -n(QjT)~ exists

[18],

and

T E 37 J

iff

FTT = s t - lim ~ n n J N ~ n=0 ~ j ( Q j T ) ~ exists.

If

T E ~(~)

and

QjT ~ 0,

(2) then

T E 3 ±j,

T ~ J~T

T = J ~ T $ F~T. J J Thus

Let T

(3)

SH(~) c 3~j' for j = 1,2,...,n. and J j are also denoted by Fj

F~

~

be an auxiliary

be the unit circle

Tj

is a copy of

in

Tn ,

mj.

vj

be a singular

~

be the

i@. J) = ~i

m = m I × ... × m n , Let

S

e

measure

on

space

which is defined

to

~ = L2(~,~,R)

S,

and

measurable

functions

llfll 2 = R(z)f(z)

and

all subsets ~

=

X jE~

~

=

X ~j, jEn

Let v

on

Tn