Antinormal composition operators on $L^2$ -space of an atomic measure space
Let $L^2(\mu)$ denotes the Hilbert space associated with a $\sigma$ -finite atomic measure $\mu$. We propose a characterization of antinormal composition operators on $L^2(\mu)$.
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| author | Chandra, H. Kumar, D. Чандра, Г. Кумар, Д. |
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| description | Let $L^2(\mu)$ denotes the Hilbert space associated with a $\sigma$ -finite atomic measure $\mu$. We propose a characterization of antinormal composition operators on $L^2(\mu)$. |
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UDC 517.9
D. Kumar, H. Chandra (Inst. Sci. Banaras Hindu Univ., Varanasi, India)
ANTINORMAL COMPOSITION OPERATORS ON \bfitL \bftwo -SPACE
OF AN ATOMIC MEASURE SPACE
АНТИНОРМАЛЬНI ОПЕРАТОРИ КОМПОЗИЦIЇ НА ПРОСТОРI \bfitL \bftwo ,
ЩО ВIДПОВIДАЄ ПРОСТОРУ З АТОМНОЮ МIРОЮ
Let L2(\mu ) denotes the Hilbert space associated with a \sigma -finite atomic measure \mu . We propose a characterization of
antinormal composition operators on L2(\mu ).
Позначимо гiльбертiв простiр, асоцiйований з \sigma -скiнченною атомною мiрою \mu , через L2(\mu ). Наведено характери-
зацiю антинормальних операторiв композицiї на L2(\mu ).
1. Introduction. A classical problem in operator theory is to determine the distance of an operator
from a given class of bounded linear operators on a Hilbert space. The distance between an operator
and the set of Hermitian, positive, compact and unitary operators have been investigated in [2,
3, 5, 7], respectively. It is therefore natural to make analogous study about the set of normal
operators. But most of the usual approximation criterion are not applicable since the set of normal
operators is not convex. In 1974 Holmes [6] investigated those operators which admit a best normal
approximation. He observed that there are operators for which its largest possible distance from the
set of normal operators can be achieved. He named such operators as antinormal and showed that
no invertible operator is antinormal and consequently, no compact operator is antinormal. Thus, the
existence of antinormal operators is infinite dimensional phenomenon. Subsequently this class has
been extensively studied by several authors in [4, 8 – 10]. In 2008 Tripathi and Lal [13] characterized
antinormal composition operators on the Hilbert space l2. In this paper we obtain a characterization
of normal and antinormal composition operators on L2(\mu ).
2. Preliminaries. In this section we give certain basic definitions and fix some notations.
Definition 2.1. Let (X,\scrS , \mu ) be a measure space. A measurable set E is called an atom if
\mu (E) \not = 0 and for each measurable subset F of E either \mu (F ) = 0 or \mu (F ) = \mu (E). A measure
space (X,\scrS , \mu ) is called atomic measure space if each measurable subset of non-zero measure
contains an atom.
A trivial example of an atomic measure space is (X,\scrS , \mu ), where X is any non-empty set, \scrS is
a \sigma -algebra and \mu is the counting measure.
Definition 2.2. An atomic measure space (X,\scrS , \mu ) is called \sigma -finite atomic measure space if
X is expressible as countable union of its atoms of finite measure.
Let (X,\scrS , \mu ) be a \sigma -finite atomic measure space. Then X =
\bigcup
n\in \BbbN An, where An are disjoint
atoms of finite measure. These atoms are unique in the sense that if X =
\bigcup
n\in \BbbN Bn, where Bn
are disjoint atoms of finite measure, then An = Bn upto a null set for each n \in \BbbN . A measurable
transformation \varphi : X \rightarrow X is called non-singular if \mu \phi - 1 is absolutely continuous with respect to
measure \mu . If a non-singular transformation on X takes some part of an atom An to a subset of
an atom Am and the remaining part of An to a subset of another atom Ar, then both part must be
null sets as image of an atom under a non-singular transformation cannot be a null set. Therefore
a non-singular measurable transformation takes atoms into atoms. A non-singular transformation \varphi
c\bigcirc D. KUMAR, H. CHANDRA, 2019
92 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
ANTINORMAL COMPOSITION OPERATORS ON L2 -SPACE OF AN ATOMIC MEASURE SPACE 93
of X into X is called injective almost everywhere (a.e.) if the inverse image of every atom under
\varphi contains at most one atom. It is called surjective a.e. if the inverse image of every atom under \varphi
contains at least one atom. If \varphi is both injective and surjective a.e., then it is called bijective a.e.
Let \BbbN and \BbbC denote the set of all positive integers and the set of all complex numbers, respec-
tively. Then for n \in \BbbN , \chi An : X \rightarrow \{ 0, 1\} be defined as
\chi An(x) =
\left\{ 1, if x \in An,
0, otherwise.
Henceforth, we take L2(\mu ) of a \sigma -finite atomic measure space (X,\scrS , \mu ), where X =
\bigcup
n\in \BbbN An
and An are disjoint atoms of finite measure. Further, each f \in L2(\mu ) is constant a.e. on an atom.
Hence, the span of the characteristic functions \{ \chi An : n \in \BbbN \} forms a dense subset of L2(\mu ). Thus,\biggl\{
KAn =
1
\mu (An)
\chi An : n \in \BbbN
\biggr\}
forms an orthonormal basis for L2(\mu ).
2.1. Composition operators. Let (X,\scrS , \mu ) be a \sigma -finite atomic measure space. A non-singular
measurable transformation \varphi induces a linear transformation C\varphi on L2(\mu ) defined by
C\varphi (f) = f \circ \varphi \forall f \in L2(\mu ).
When C\varphi is bounded, it is called composition operator. A necessary and sufficient condition for
boundedness of C\varphi is given below.
Theorem 2.1 [11]. A composition transformation C\phi is bounded on L2(\mu ) if and only if there
exists a positive real number M > 0 such that \mu (\varphi - 1(E)) \leq M\mu (E) \forall E \in \scrS .
Theorem 2.2 [11]. Let C\varphi be a composition operator. Then C\varphi is normal if and only if the
range of C\varphi is dense in L2(\mu ) and f\varphi \circ \varphi = f\varphi a.e., where f\varphi is a Radon – Nikodym derivative of
\mu \varphi - 1 w.r.t. \mu .
In 1983 Singh and Veluchamy computed the adjoint of a composition operator on L2(\mu ) and
gave following characterization for an operator to be a composition operator. Further, they obtained
adjoint of the composition operator as follows.
Theorem 2.3 [12]. Let C\varphi be a composition operator. Then
(C\ast
\varphi f)(An) =
1
\mu (An)
\int
\varphi - 1(An)
f d\mu
a.e. for f \in L2(\mu ) and for every atom An.
Moreover, they gave following characterization for an operator to be a composition operator.
Theorem 2.4 [12]. Let T be a bounded linear operator on L2(\mu ). Then T is a composition
operator if and only if the set \{ KAn : n \in \BbbN \} is invariant under T \ast . In this case \varphi is determined by
T \ast (KAn) = K\varphi (An).
2.2. Antinormal operators on Hilbert space. Suppose H is a complex Hilbert space and B(H)
is the algebra of all bounded linear operators on H. Further, for T \in B(H), let N(T ) and R(T )
respectively denote the null space and the range space of T.
Definition 2.3 [1]. An operator T \in B(H) is said to be Fredholm operator if dimension of
N(T ) and the dimension of the quotient space H/R(T ) are both finite.
Equivalently, T is Fredholm if both N(T ) and N(T \ast ) are finite dimensional.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
94 D. KUMAR, H. CHANDRA
Definition 2.4. Essential spectrum of an operator T \in B(H) is defined as \sigma e(T ) = \{ \alpha \in \BbbC :
T - \alpha I is not Fredholm\} .
Since every invertible operator is Fredholm operator, \sigma e(T ) \subseteq \sigma (T ).
Definition 2.5. Minimum modulus of an operator T \in B(H) is defined as m(T ) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \| Tx\| :
\| x\| = 1\} .
Definition 2.6. Essential minimum modulus of an operator T \in B(H) is defined as me(T ) =
= \mathrm{i}\mathrm{n}\mathrm{f}\{ \alpha \geq 0 : \alpha \in \sigma e(| T | )\} , where | T | = (T \ast T )1/2.
Definition 2.7. An operator T \in B(H) is said to be antinormal if d(T,\scrN ) = \mathrm{i}\mathrm{n}\mathrm{f}N\in \scrN \| T -
- N\| = \| T\| , where \scrN is the class of all normal operators in B(H).
Remark 2.1. An operator T \in B(H) is antinormal if and only if its adjoint T \ast is antinormal.
Definition 2.8. For an operator T in B(H), index of T is defined as
\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) =
\left\{
\mathrm{d}\mathrm{i}\mathrm{m}(N(T )) - \mathrm{d}\mathrm{i}\mathrm{m}(N(T \ast )), if \mathrm{d}\mathrm{i}\mathrm{m}(N(T )) or
\mathrm{d}\mathrm{i}\mathrm{m}(N(T \ast )) < \infty ,
0, otherwise.
Remark 2.2. Observe that \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) = - \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T \ast ).
The following results will be used in the later part of the paper.
Theorem 2.5 [8]. Let T \in B(H).
(i) If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) = 0, then d(T,\scrN ) \leq \| T\| - m(T )
2
.
(ii) If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) < 0, then me(T ) \leq d(T,\scrN ) \leq \| T\| +me(T )
2
.
Remark 2.3. If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) = 0, then T can not be antinormal.
Theorem 2.6 [8]. Let T \in B(H) with \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(T ) < 0. Then following conditions are equivalent:
(i) T is antinormal;
(ii) me(T ) = \| T\| ;
(iii) d(T,\scrU ) = 1 + \| T\| , where \scrU is the class of all unitary operators in B(H);
(iv) T = \alpha W (1 - K) for some \alpha > 0, isometry W and positive compact contraction K.
3. Antinormal composition operators on \bfitL \bftwo (\bfitmu ). We begin with a characterization of normal
composition operators on L2(\mu ).
Theorem 3.1. C\varphi is an injective if and only if \varphi is surjective a.e.
Proof. Suppose that \varphi is surjective a.e. Therefore, \varphi (\varphi - 1(An)) = An \forall n \in \BbbN . Suppose
f, g \in L2(\mu ) be such that
C\varphi f = C\varphi g.
Then
f(\varphi (\varphi - 1(An))) = g(\varphi (\varphi - 1(An))) \forall n \in \BbbN
\Rightarrow f(An) = g(An) \forall n \in \BbbN .
Thus, f = g. Therefore, C\varphi is injective. Conversely, suppose that C\varphi is injective. It follows that for
each n \in \BbbN
C\varphi (\chi An) \not = 0,
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ANTINORMAL COMPOSITION OPERATORS ON L2 -SPACE OF AN ATOMIC MEASURE SPACE 95
\chi \varphi - 1(An) \not = 0.
Hence, \mu (\varphi - 1(An)) \not = 0 for each n \in \BbbN . Consequently \varphi is surjective a.e.
Theorem 3.2. C\varphi is surjective if and only if \varphi is injective a.e.
Proof. Suppose \varphi (An) = \varphi (Am) for m,n \in \BbbN . Since \chi An \in L2(\mu ) and C\varphi is surjective, there
is a function f \in L2(\mu ) such that C\varphi f = \chi An . Therefore,
f(\varphi (An)) = 1 and f(\varphi (Am)) = \delta n,m,
where \delta n,m is Kronecker delta. In view of the fact that \varphi (An) = \varphi (Am) we have \delta n,m = 1.
Hence, n = m. Conversely, suppose that \varphi is injective a.e. Let f \in L2(\mu ). Define a function g :
X \rightarrow \BbbC as follows. For each n \in \BbbN
g(An) =
\left\{ f(Am), if \varphi (Am) = An for some n \in \BbbN ,
0, otherwise.
Function g is well defined as \varphi is injective a.e. Also
\| g\| 2 =
\int
X
| g| 2 d\mu =
\sum
n\in \BbbN
\int
An
| g| 2 d\mu =
=
\sum
n\in \BbbN
\int
\varphi - 1(An)
| f | 2 d\mu =
\int
X
| f | 2 d\mu = \| f\| 2 < \infty .
Therefore, g \in L2(\mu ). Now it is easy to see that C\varphi g = f. Hence, C\varphi is surjective.
Theorem 3.3. C\varphi is normal if and only if \varphi is bijective a.e.
Proof. It is immediate from Theorems 2.2, 3.1 and 3.2.
Remark 3.1. If \varphi is bijective a.e., then C\varphi is a non-zero normal operator. Hence it is not
antinormal.
Theorem 3.4. Let \varphi : X \rightarrow X be a non-singular transformation such that \varphi is injective a.e.
but not surjective a.e. Then C\varphi is antinormal.
Proof. Since \varphi is not surjective a.e., there exists an atom An0 such that \mu (\varphi - 1(An0)) = 0.
Therefore,
C\varphi KAn0
= K\varphi - 1(An0 )
= 0.
Consequently C\varphi is not injective.
Let f =
\sum
n\in \BbbN
f | AnKAn \in L2\mu . Then
C\ast
\varphi f =
\sum
n\in \BbbN
f | AnK\varphi (An).
Since \varphi is injective, \varphi (Am) = \varphi (An) implies Am = An. Thus, C\ast
\varphi f = 0 asserts f = 0. Hence, C\ast
\varphi
is injective. Therefore, \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) < 0. Now let f =
\sum
n\in \BbbN
f | AnKAn \in L2(\mu ). We get
(C\varphi C
\ast
\varphi - \alpha I)f = (1 - \alpha )f for each \alpha \in \BbbC . (3.1)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
96 D. KUMAR, H. CHANDRA
From equation (3.1) it follows that C\varphi C
\ast
\varphi - \alpha I is invertible whenever \alpha \not = 1. Consequently C\varphi C
\ast
\varphi -
- \alpha I is Fredholm for each \alpha \not = 1. Thus,
\surd
\alpha /\in \sigma e(| C\ast
\varphi | ) for each \alpha \not = 1. Again, by above equation,
\mathrm{d}\mathrm{i}\mathrm{m} \mathrm{k}\mathrm{e}\mathrm{r}(C\varphi C
\ast
\varphi - I) is infinite. Therefore, 1 \in \sigma e(| C\ast
\varphi | ). Thus, me(C
\ast
\varphi ) = 1 = \| C\varphi \| . Hence, C\ast
\varphi
is antinormal by Theorem 2.6. Since adjoint of an antinormal operator is antinormal, therefore C\varphi is
antinormal.
The following result gives a necessary and sufficient condition for antinormality of C\varphi if \varphi is
surjective a.e. but not injective a.e.
Theorem 3.5. Suppose \varphi is surjective a.e. but is not injective a.e. Then C\varphi is antinormal if and
only if the following conditions hold:
(a) for each 0 \leq \alpha < \| C\varphi \| 2,
\mu \varphi - 1(An)
\mu (An)
\not = \alpha except for finitely many n \in \BbbN ;
(b) \| C\varphi \| 2 =
\mu \varphi - 1(An)
\mu (An)
for infinitely many n \in \BbbN .
Proof. Let f \in L2(\mu ) be such that C\varphi (f) = 0. Then (f \circ \varphi )(An) = 0 \forall n \in \BbbN . Now surjectivity
of \varphi implies f = 0. Further, since \varphi is not injective a.e., there exist m, n \in \BbbN with m \not = n such
that \varphi (Am) = \varphi (An). Now put f = KAn and g = KAm . It is easy to see that f, g \in L2(\mu ) and
f \not = g but C\ast
\varphi (f) = C\ast
\varphi (g). Therefore, C\ast
\varphi is not injective. Let f =
\sum
n\in \BbbN
f | AnKAn \in L2(\mu ) and
\alpha be a real number. Then
(C\ast
\varphi C\varphi - \alpha I)f =
\sum
n\in \BbbN
\biggl(
\mu \varphi - 1(An)
\mu (An)
- \alpha
\biggr)
f | AnKAn . (3.2)
It follows by condition (a) and equation (3.2) that C\ast
\varphi C\varphi - \alpha I is Fredholm for each 0 \leq \alpha < \| C\varphi \| 2.
Therefore,
\surd
\alpha /\in \sigma e(| C\varphi | ) for each 0 \leq \alpha < \| C\varphi \| 2. Condition (b) together with equation (3.2)
asserts that C\ast
\varphi C\varphi - \alpha I is not Fredholm for \alpha = \| C\phi \| 2. Hence, \| C\phi \| \in \sigma e(| C\varphi | ) Thus, me(C\varphi ) =
= \| C\varphi \| . Consequently C\varphi is antinormal by Theorem 2.6. Conversely, if either of the conditions
is not true, then we claim that C\varphi is not antinormal. If condition (a) fails, then there exists an \alpha 0
with 0 \leq \surd
\alpha 0 < \| C\varphi \| 2, such that
\surd
\alpha 0 \in \sigma e(| C\varphi | ). Therefore, me(C\varphi ) \leq \surd
\alpha 0 < \| C\varphi \| and so
C\varphi is not antinormal. Now suppose condition (b) fails. Then, by equation (3.2), \| C\varphi \| /\in \sigma e(| C\varphi | ).
Therefore, me(C\varphi ) < \| C\varphi \| . Thus, C\varphi is not antinormal in either case.
Following theorem characterize antinormal composition operator when \phi is neither injective nor
surjective.
Theorem 3.6. Suppose \varphi is neither injective nor surjective a.e.
(i) If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) \geq 0, C\varphi is not antinormal.
(ii) If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) < 0, C\varphi is antinormal if and only if following conditions hold:
(a) for each 0 \leq \alpha < \| C\varphi \| 2,
\mu \varphi - 1(An)
\mu (An)
\not = \alpha except for finitely many n \in \BbbN ;
(b) \| C\varphi \| 2 =
\mu \varphi - 1(An)
\mu (An)
for infinitely many n \in \BbbN .
Proof. Suppose that \varphi is neither injective a.e. nor surjective a.e. We split the proof of (i) into
two case.
Case I. If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) = 0, then it is not antinormal by Remark 2.3.
Case II. If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) > 0, then \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\ast
\varphi ) < 0. Therefore \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}C\ast
\varphi is finite. But by [12]
we have
\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}C\ast
\varphi =
\sum
n\in \BbbN
(\alpha n - 1),
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
ANTINORMAL COMPOSITION OPERATORS ON L2 -SPACE OF AN ATOMIC MEASURE SPACE 97
where \alpha n denotes the number of atoms in \varphi - 1(An) for each n \in \BbbN . Consequently \alpha n = 1 for all
but finitely many n \in \BbbN . Now by
(C\varphi C
\ast
\varphi - I)f =
\sum
n\in \BbbN
f | AnK\varphi - 1(\varphi (An)) -
\sum
n\in \BbbN
f | AnKAn .
Hence, K\varphi - 1(An) \in \mathrm{k}\mathrm{e}\mathrm{r}(C\varphi C
\ast
\varphi - I) for all but finitely many n \in \BbbN . Therefore, \mathrm{k}\mathrm{e}\mathrm{r}((C\varphi C
\ast
\varphi - I))
is infinite dimensional. Thus, 1 \in \sigma e(| C\ast
\varphi | ). Therefore, me(C
\ast
\varphi ) \leq 1 < \| C\varphi \| . Hence, C\varphi is
antinormal.
(ii) If \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) < 0, then result follows using arguments used in Theorem 3.5.
Now we present some examples to illustrate obtained results.
Example 3.1. Let X = [0,\infty ) with atoms An = [n - 1, n] and \mu (An) = 1 for each n \in \BbbN .
Let \varphi (x) = 2x. Let E be a measurable set such that \mu (E) = 0. Then \mu (\varphi - 1(E)) = \mu
\biggl(
1
2
E
\biggr)
= 0.
Hence, \varphi is non singular. It is easy to see that \varphi is injective but not surjective since \mu (\varphi - 1(An)) = 0
for each n \in \BbbN . Hence, by Theorem 3.4, the composition operator C\varphi induced by \varphi is antinormal.
Example 3.2. Let X = N, \mu = counting measure and An = \{ n\} for each n \in \BbbN . Define a
function \varphi on \BbbN such that
\varphi (n) =
\left\{
1, if n = 1,
2, if n = 2, 3,
3, if n = 4,
n+ 3
2
, if n (\geq 5) is odd,
n+ 2
2
, if n (\geq 6) is even.
It is easy to see that \varphi is surjective but not injective. Also \| C\varphi \| 2 = \mathrm{s}\mathrm{u}\mathrm{p}\{ | \varphi - 1(n)| : n \in \BbbN \} = 2,
where | \varphi - 1(n)| denotes cardinality of the set \varphi - 1(n) for each n \in \BbbN . Further,
\mu \varphi - 1(An)
\mu (An)
=
\left\{ 1, if n = 1, 3,
2, otherwise.
Thus, both the conditions of Theorem 3.5 are satisfied. Therefore, C\varphi is antinormal.
Example 3.3. Let X = N and \mu = counting measure. Define a function \varphi on \BbbN such that
\varphi (n) =
\left\{ 1, if n = 1, 2,
n+ 2, otherwise.
It is easy to see that \varphi is neither surjective but not injective. \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) = \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}(C\varphi ) -
\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}(C\ast
\varphi ) = 3 - 1 = 2 > 0. Therefore, C\varphi is not antinormal by Theorem 3.6 (a).
Example 3.4. Let X = N, \mu = counting measure and An = \{ n\} for each n \in \BbbN . Define a
function \varphi on \BbbN such that
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
98 D. KUMAR, H. CHANDRA
\varphi (n) =
\left\{
1, if n = 1,
2, if n = 2, 3,
4 +m - 1, if n = 2m+ 2, 2m+ 3, where m \in \BbbN .
It is easy to see that \varphi is neither surjective but not injective. Since \mathrm{k}\mathrm{e}\mathrm{r}(C\varphi ) is one dimensional and
\mathrm{k}\mathrm{e}\mathrm{r}(C\ast
\varphi ) is infinite dimensional, hence \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}(C\varphi ) < 0. Also \| C\varphi \| 2 = 2. Further,
\mu \varphi - 1(An)
\mu (An)
=
\left\{
1, if n = 1,
0, if n = 3,
2, otherwise.
Thus, both the conditions posed in Theorem 3.6 (b) are satisfied. Therefore, C\varphi is antinormal.
References
1. Abramovich Y. A., Aliprantis C. D. An invitation to operator theory // Grad. Stud. Math. – Amer. Math. Soc., 2002.
2. AFM ter Elst. Antinormal operators // Acta Sci. Math. – 1990. – 54. – P. 151 – 158.
3. Fan K., Hoffman A. Some metric inequalities in the space of matrices // Proc. Amer. Math. Soc. – 1955. – 6. –
P. 111 – 116.
4. Fujii M., Nakamoto R. Antinormal operators and theorems of Izumino // Math. Jap. – 1979. – 24. – P. 41 – 44.
5. Halmos P. R. Positive approximants of operators // Indiana Univ. Math. J. – 1971/72. – 21. – P. 951 – 960.
6. Holmes R. B. Best approximation by normal operators // J. Approxim. Theory. – 1974. – 12. – P. 412 – 417.
7. Holmes R. B., Kripke B.R. Best approximation by compact operators // Indiana Univ. Math. J. – 1971. – 21. –
P. 253 – 255.
8. Izumino S. Inequalities on normal and antinormal operators // Math. Jap. – 1978. – 23. – P. 211 – 215.
9. Izumino S. Inequalities on nilpotent operators // Math. Jap. – 1979. – 24. – P. 31 – 34.
10. Rogers D. D. On proximal sets of normal operators // Proc. Amer. Math. Soc. – 1976. – 61. – P. 44 – 48.
11. Singh R. K., Manhas J. S. Composition operators on function spaces. – North-Holland, New York, 1993.
12. Singh R. K., Veluchamy T. Atomic measure spaces and essentially normal composition operators // Bull. Austral.
Math. Soc. – 1983. – 27. – P. 259 – 267.
13. Tripathi G. P., Lal N. Antinormal composition operators on l2 // Tamkang J. Math. – 2008. – 39. – P. 347 – 352.
Received 07.04.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
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| id | umjimathkievua-article-1421 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:01Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ad/f6625bd50075899daa9c765cf8d2aaad.pdf |
| spelling | umjimathkievua-article-14212019-12-05T08:54:16Z Antinormal composition operators on $L^2$ -space of an atomic measure space Антинормальнi оператори композицiї на просторi $L^2$, що вiдповiдає простору з атомною мiрою Chandra, H. Kumar, D. Чандра, Г. Кумар, Д. Let $L^2(\mu)$ denotes the Hilbert space associated with a $\sigma$ -finite atomic measure $\mu$. We propose a characterization of antinormal composition operators on $L^2(\mu)$. Позначимо гiльбертiв простiр, асоцiйований з \sigma -скiнченною атомною мiрою $\mu$, через $L^2(\mu)$. Наведено характеризацiю антинормальних операторiв композицiї на $L^2(\mu)$. Institute of Mathematics, NAS of Ukraine 2019-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1421 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 1 (2019); 92-98 Український математичний журнал; Том 71 № 1 (2019); 92-98 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1421/405 Copyright (c) 2019 Chandra H.; Kumar D. |
| spellingShingle | Chandra, H. Kumar, D. Чандра, Г. Кумар, Д. Antinormal composition operators on $L^2$ -space of an atomic measure space |
| title | Antinormal composition operators on $L^2$ -space of an atomic measure
space |
| title_alt | Антинормальнi оператори композицiї на просторi $L^2$,
що вiдповiдає простору з атомною мiрою |
| title_full | Antinormal composition operators on $L^2$ -space of an atomic measure
space |
| title_fullStr | Antinormal composition operators on $L^2$ -space of an atomic measure
space |
| title_full_unstemmed | Antinormal composition operators on $L^2$ -space of an atomic measure
space |
| title_short | Antinormal composition operators on $L^2$ -space of an atomic measure
space |
| title_sort | antinormal composition operators on $l^2$ -space of an atomic measure
space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1421 |
| work_keys_str_mv | AT chandrah antinormalcompositionoperatorsonl2spaceofanatomicmeasurespace AT kumard antinormalcompositionoperatorsonl2spaceofanatomicmeasurespace AT čandrag antinormalcompositionoperatorsonl2spaceofanatomicmeasurespace AT kumard antinormalcompositionoperatorsonl2spaceofanatomicmeasurespace AT chandrah antinormalʹnioperatorikompoziciínaprostoril2ŝovidpovidaêprostoruzatomnoûmiroû AT kumard antinormalʹnioperatorikompoziciínaprostoril2ŝovidpovidaêprostoruzatomnoûmiroû AT čandrag antinormalʹnioperatorikompoziciínaprostoril2ŝovidpovidaêprostoruzatomnoûmiroû AT kumard antinormalʹnioperatorikompoziciínaprostoril2ŝovidpovidaêprostoruzatomnoûmiroû |