Fuglede – Putnam type theorems for extension of $M$-hyponormal operators
UDC 517.9 We consider $k$-quasi-$M$-hyponormal operator $T \in B(\mathcal{H})$ suchthat $TX = XS$ for some $X \in B(\mathcal{K},\mathcal{H})$ and prove the Fuglede–Putnam type theorem when adjoint of $S \in B(\mathcal{K})$ is $k$-quasi-$M$-hyponormal or dominant operators.We also show that two quasi...
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| author | Mecheri , S. Prasad , T. Mecheri , S. Prasad , T. |
| author_facet | Mecheri , S. Prasad , T. Mecheri , S. Prasad , T. |
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| description | UDC 517.9
We consider $k$-quasi-$M$-hyponormal operator $T \in B(\mathcal{H})$ suchthat $TX = XS$ for some $X \in B(\mathcal{K},\mathcal{H})$ and prove the Fuglede–Putnam type theorem when adjoint of $S \in B(\mathcal{K})$ is $k$-quasi-$M$-hyponormal or dominant operators.We also show that two quasisimilar $k$-quasi-$M$-hyponormal operators have equal essential spectra.
 
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| doi_str_mv | 10.37863/umzh.v74i1.2355 |
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DOI: 10.37863/umzh.v74i1.2355
UDC 517.9
S. Mecheri (Mohamed El Bachir El Ibrahimi Univ., Bordj Bou Arreridj, Algeria),
T. Prasad (Univ. Calicut, Kerala, India)
FUGLEDE – PUTNAM TYPE THEOREMS
FOR EXTENSION OF \bfitM -HYPONORMAL OPERATORS
ТЕОРЕМИ ТИПУ ФУГЛЕДЕ – ПУТНАМА
ДЛЯ РОЗШИРЕНЬ \bfitM -ГIПОНОРМАЛЬНИХ ОПЕРАТОРIВ
We consider k-quasi-M-hyponormal operator T \in B(\scrH ) such that TX = XS for some X \in B(\scrK ,\scrH ) and prove the
Fuglede – Putnam type theorem when adjoint of S \in B(\scrK ) is k-quasi-M-hyponormal or dominant operators. We also
show that two quasisimilar k-quasi-M-hyponormal operators have equal essential spectra.
Розглянуто k-квазi-M-гiпонормальний оператор T \in B(\scrH ) такий, що TX = XS для деякого X \in B(\scrK ,\scrH ), та
доведено теорему типу Фугледе – Путнама, коли спряженим до S \in B(\scrK ) є або k-квазi-M-гiпонормальний, або
домiнуючий оператор. Також показано, що два квазiподiбнi k-квазi-M-гiпонормальнi оператори мають однаковi
суттєвi спектри.
1. Introduction. Let \scrH and \scrK be separable complex Hilbert spaces, and let B(\scrH ,\scrK ) denote the
algebra of all bounded linear operators from \scrH to \scrK (We also write B(\scrH ) = B(\scrH ,\scrH ).) Throughout
this paper, the range and the null space of an operator T will be denoted by ran(T ) and \mathrm{k}\mathrm{e}\mathrm{r}(T ),
respectively. Let \scrM and \scrM \bot be the norm closure and the orthogonal complement of the subspace
\scrM of \scrH . The classical Fuglede – Putnam theorem [4] (Problem 152) asserts that if T \in B(\scrH )
and S \in B(\scrK ) are normal operators such that TX = XS for some operators X \in B(\scrK ,\scrH ), then
T \ast X = XS\ast . The references [2, 6, 9, 10, 17 – 19] are among the various extensions of this celebrated
theorem for nonnormal operators. According to [17], an operator T \in \scrH is dominant if
\mathrm{r}\mathrm{a}\mathrm{n}(T - \lambda I) \subseteq \mathrm{r}\mathrm{a}\mathrm{n}(T - \lambda I)\ast for all \lambda \in \BbbC .
From [1], it is seen that this condition is equivalent to the existence of a positive constant M\lambda such
that
(T - \lambda I)(T - \lambda I)\ast \leq M2
\lambda (T - \lambda I)\ast (T - \lambda I)
for each \lambda \in \BbbC . An operator T is called M-hyponormal if there is a constant M such that M\lambda \leq M
for all \lambda \in \BbbC . If M = 1, T is hyponormal. We have the following inclusion relations:
\{ hyponormal\} \subseteq \{ M -hyponormal\} \subseteq \{ dominant\} .
Mecheri [5] introduced k-quasi-M-hyponormal operators as follows. An operator T is k-quasi-M-
hyponormal if there exists a real positive number M such that
T \ast k\bigl( (T - \lambda I)(T - \lambda I)\ast
\bigr)
T k \leq T \ast k(M2
\bigl(
T - \lambda I)\ast (T - \lambda I)
\bigr)
T k
for all \lambda \in \BbbC , where k is a natural number. Evidently,
\{ M -hyponormal\} \subseteq \{ k-quasi-M -hyponormal\} .
c\bigcirc S. MECHERI, T. PRASAD, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1 89
90 S. MECHERI, T. PRASAD
For T \in B(\scrH ) and S \in B(\scrK ), we say that FP -theorem holds for the pair (T, S) if TX = XS
implies T \ast X = XS\ast , \mathrm{r}\mathrm{a}\mathrm{n}(X) reduces T, and \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot reduces S, the restrictions T |
ran(X)
and
S| ker(X)\bot are unitarily equivalent normal operators for all X \in B(\scrK ,\scrH ). We say that an operator
S is quasiaffine transform of an operator T if there exists an injective operator X with dense range
such that TX = XS. Two operators T \in B(\scrH ) and S \in B(\scrH ) are quasisimilar if there exist
quasiaffinities X \in B(\scrH ,\scrK ) and Y \in B(\scrK ,\scrH ) such that XT = SX and Y S = TY. In general
quasisimilarity need not preserve the spectrum and essential spectrum. However, in special classes
of operators quasisimilarity preserves spectra. For instance, it is well-known that two quasisimilar
hyponormal operators have equal spectrum and equal essential spectrum.
Recall that an operator T \in B(\scrH ) is k-quasihyponormal if T \ast k(T \ast T - TT \ast )T k \geq 0, where
k is a positive integer and an operator T \in B(\scrH ) is said to be (p, k)-quasihyponormal operators if
T \ast k\bigl( (T \ast T )p - (TT \ast )p
\bigr)
T k \geq 0, where k is a positive integer and 0 < p \leq 1 [3, 19]. Recently,
Tanahashi, Patel and Uchiyama [19] found some extensions of Fuglede – Putnam theorems involving
(p, k)-quasihyponormal, dominant, and spectral operators.
Recall [8] that an operator T \in B(\scrH ) is said to have the single-valued extension property (SVEP)
if for every open subset D of \BbbC and any analytic function f : D \rightarrow \scrH such that (T - \lambda )f(\lambda ) \equiv 0 on
D, it results f(\lambda ) \equiv 0 on D. We say that a Hilbert space operator satisfies Bishop property (\beta ) if, for
every open subset D of \BbbC and every sequence fn : D - \rightarrow \scrH of analytic functions with (T - \lambda )fn(\lambda )
converges uniformly to 0 in norm on compact subsets of D, fn(\lambda ) converges uniformly to 0 in norm
on compact subsets of D. It is well-known that
Bishop property(\beta ) \Rightarrow SVEP
(see [8] for more information). Mecheri [5] proved that k-quasi-M-hyponormal operators satisfies
Bishop property (\beta ). Recently, some spectral properties of k-quasi-M-hyponormal operators has
been studied by Zuo and Mecheri [22]. In the present note, we seek some extensions of Fuglede –
Putnam type theorems involving k-quasi-M-hyponormal operator and dominant operators. Let U
be an open set in \BbbC . Stampfli [16] investigated the equation (T - \lambda I)f(\lambda ) \equiv x for some non-zero
x \in \scrH and f : U \rightarrow \scrH in an effort to discover necessary and or sufficient condition for analyticity
of f when T is a dominant operator. In this paper, we show that if T \in B(\scrH ) be k-quasi-M-
hyponormal, if 0 /\in \delta \subseteq \BbbC be closed, and if there exists a bounded function f : \BbbC \setminus \delta \rightarrow \scrH such
that (T - \lambda I)f(\lambda ) \equiv x for some non-zero x \in H, then f is analytic at every non zero point and
hence f has analytic extension everywere on \BbbC \setminus \delta . In Section 3, we show that if T, S \in B(\scrH ) are
quasisimilar k-quasi-M-hyponormal operators, then they have equal spectrum.
2. Fuglede – Putnam type theorem. Throughout this paper we would like to present some
known results as propositions which will be used in the sequel.
Proposition 2.1 [5]. Let T be k-quasi-M-hyponormal operator, \mathrm{r}\mathrm{a}\mathrm{n}(T k) be not dense and
T =
\Biggl(
T1 T2
0 T3
\Biggr)
on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(T k)\oplus \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k).
Then T1 = T |
ran(Tk)
is M-hyponormal, T k
3 = 0 and \sigma (T ) = \sigma (T1) \cup \{ 0\} .
Proposition 2.2 [15]. Let T \in B(\scrH ) and let S \in B(\scrK ). Then the following assertions are
equivalent:
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
FUGLEDE – PUTNAM TYPE THEOREMS FOR EXTENSION OF M-HYPONORMAL OPERATORS 91
(i) If TX = XS where X \in B(\scrK ,\scrH ), then T \ast X = XS\ast .
(ii) If TX = XS where X \in B(\scrK ,\scrH ), then \mathrm{r}\mathrm{a}\mathrm{n}(X) reduces T, and \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot reduces S, the
restrictions T |
ran(X)
and S| ker(X)\bot are normal.
Proposition 2.3 [10]. Let T and S be M-hyponormal operators and TX = XS\ast . Then
(i) \mathrm{r}\mathrm{a}\mathrm{n}(X) reduces T and \mathrm{k}\mathrm{e}\mathrm{r}(X) reduces S.
(ii) T |
ran(X)
and S\ast | ker(X)\bot are unitarily equivalent normal operators.
It is well-known that a normal part of hyponormal is reducing. This result remains true for
dominant operators.
Proposition 2.4 [14, 17, 21]. Let T \in B(\scrH ) be dominant and \scrM be an invariant subspace of
T. Then:
(i) The restriction T | \scrM is dominant.
(ii) If the restriction T | \scrM is normal, then \scrM reduces T.
In the following lemma we prove, a normal part of a k-quasi-M-hyponormal operator is reducing.
Lemma 2.1. If the restriction T | \scrM of the k-quasi-M-hyponormal operator T \in \scrB (\scrH ) to an
invariant subspace \scrM is injective and normal, then \scrM reduces T.
Proof. Let T be k-quasi-M-hyponormal and T1 = T | \scrM is injective and normal. Decompose T
on \scrH = \scrM \oplus \scrM \bot as follows:
T =
\Biggl(
T1 T2
0 T3
\Biggr)
.
The following inclusion relation holds by the k-quasi-M-hyponormality of T and Theorem 1 of [1]:
\mathrm{r}\mathrm{a}\mathrm{n}(T \ast k(T - \lambda I)) \subset \mathrm{r}\mathrm{a}\mathrm{n}(T \ast k(T \ast - \lambda I)) \subset \mathrm{r}\mathrm{a}\mathrm{n}(T \ast - \lambda I)
for \lambda \in \BbbC . Then, for any arbitrary vector y \in \scrM \bot , T \ast k
1 T2y = (T \ast
1 - \lambda )u\lambda for some u\lambda \in \scrM .
Choose v\lambda such that (T \ast
1 - \lambda I)u\lambda = (T1 - \lambda I)v\lambda . It follows that T \ast k
1 T2y = (T1 - \lambda )v\lambda and so
T \ast k
1 T2y \in \cap
\lambda \in \BbbC
\mathrm{r}\mathrm{a}\mathrm{n}(T1 - \lambda I).
Then, by [11] (Theorem 1), T \ast k
1 T2y = 0 and hence T2y = 0. Therefore, T2 = 0.
Remark 2.1. The condition T | \scrM is injective in Lemma 2.1 is indispensable because \mathrm{k}\mathrm{e}\mathrm{r}(T ) for
k-quasi-M-hyponormal operator T is not always reducing.
In [19], the authors considered the situation S and T \ast are (p, k)-quasihyponormal operators and
proved Fuglede – Putnam theorem for (S, T ) if either S or T is injective. Now we study Fuglede –
Putnam theorem for the case that T and S\ast are k-quasi-M-hyponormal operators with the condition
that either T or S\ast is injective.
Theorem 2.1. Let T \in B(\scrH ) and S\ast \in B(\scrK ) be k-quasi-M-hyponormal operators. If either
T or S\ast is injective, then Fuglede – Putnam theorem holds for (T, S).
Proof. Suppose T and S\ast are k-quasi-M-hyponormal operators and TX = XS for any operator
X \in B(\scrK ,\scrH ). Since \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{X}) is invariant under T and \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot is invariant under S\ast , we decompose
T, S and X into
T =
\Biggl(
T1 T2
0 T3
\Biggr)
on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(X)\oplus \mathrm{r}\mathrm{a}\mathrm{n}(X)
\bot
,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
92 S. MECHERI, T. PRASAD
S =
\Biggl(
S1 0
S2 S3
\Biggr)
on \scrK = \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(X),
and
X =
\biggl(
X1 0
0 0
\biggr)
on \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(X) \rightarrow \mathrm{r}\mathrm{a}\mathrm{n}(X)\oplus \mathrm{r}\mathrm{a}\mathrm{n}(X)
\bot
,
where T1 and S\ast
1 are k-quasi-M-hyponormal operators by Proposition 2.1, and
X1 : \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \rightarrow \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{X})
is injective with dense range.
From TX = XS, we have
T1X1 = X1S1. (2.1)
First consider the case where T is injective. Clearly, T1 is injective. It is not difficult to show
from (2.1) that S1 is injective or equivalently, \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{S}\ast 1) is dense. Incidently, S\ast
1 turns out to be a M-
hyponormal operator. In particular, \mathrm{k}\mathrm{e}\mathrm{r}(S\ast
1) \subset \mathrm{k}\mathrm{e}\mathrm{r}(S1) and hence \mathrm{k}\mathrm{e}\mathrm{r}(S\ast
1) = 0. From (2.1), it is easy
to see that T \ast
1 is injective, thereby T1 is M-hyponormal. Next consider the case that S\ast is injective.
Then S\ast
1 is injective and so T \ast
1 is injective by (2.1). Obviously, T1 is an injective M-hyponormal
operator, and, by (2.1), S1 is injective. Therefore, S\ast
1 is M-hyponormal. Ultimately, if either T or
S\ast is injective, then T1 and S\ast
1 are both M-hyponormal operators. Then, by Propositions 2.2 and
2.3 and (2.1), we obtain
T \ast
1X1 = X1S
\ast
1
and T1, S1 are normal operators. Since T1 and S1 are injective, T2 = S2 = 0 by Lemma 2.1.
Hence,
T \ast X = T \ast
1X1 = X1S
\ast
1 = XS\ast .
The rest of the proof follows from Proposition 2.2.
Corollary 2.1. Let T \in B(\scrH ) and S\ast \in B(\scrK ) be k-quasi-M-hyponormal operators with reduc-
ing kernels. Then Fuglede – Putnam theorem holds for (T, S).
Proof. By hypothesis, we can write T = T1 \oplus T2 on \scrH = \scrH 1 \oplus \scrH 2 and S = S\ast
1 \oplus S\ast
2 with
respect to \scrK = \scrK 1 \oplus \scrK 2, where T1 and S\ast
1 are normal parts and T2 and S2 are pure parts. Let
X =
\Biggl(
X1 X2
X3 X4
\Biggr)
on \scrK 1 \oplus \scrK 2 \rightarrow \scrH 1 \oplus \scrH 2.
From TX = XS, we have \Biggl(
T1X1 T1X2
T2X3 T2X4
\Biggr)
=
\Biggl(
X1S1 X2S2
X3S1 X4S2
\Biggr)
.
The underlying kernel conditions ensures of T2 and S\ast
2 are injective. The operator T2 is injective
k-quasi-M-hyponormal and S1 normal. From the above matrix relation, we obtain T2X3 = X3S1.
Then by applying Theorem 2.1, we get T \ast
2X3 = X3S
\ast
1 , ran(X3) reduces T2 and T2| ran(X3)
is normal
and so X3 = 0. In a similar manner we have X2 = 0 from T1X2 = X2S2 and X4 = 0 from
T2X4 = X4S2. Since T1 and S1 are normal and since T1X1 = X1S1, required result follows from
classical Fuglede – Putnam theorem and Proposition 2.2.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
FUGLEDE – PUTNAM TYPE THEOREMS FOR EXTENSION OF M-HYPONORMAL OPERATORS 93
Proposition 2.5 [21]. If T \ast \in B(\scrH ) is M-hyponormal, S \in B(\scrK ) is dominant, and XT = SX
for X \in B(\scrH ,\scrK ), then XT \ast = S\ast X.
Now we consider the situation that where T is a k-quasi-M-hyponormal operator and S\ast is a
dominant operator.
Theorem 2.2. Let T \in B(\scrH ) be k-quasi-M-hyponormal and S\ast \in B(\scrK ) be dominant. If
either T or S\ast is injective, then Fuglede – Putnam theorem holds for (T, S).
Proof. Suppose that T \in B(\scrH ) is k-quasi-M-hyponormal and S\ast \in B(\scrK ) is dominant such
that TX = XS for X \in B(\scrK ,\scrH ). Since \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{X}) is invariant under T and \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot is invariant
under S\ast , we can write T, S and X as follows:
T =
\Biggl(
T1 T2
0 T3
\Biggr)
on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{X})\oplus \mathrm{r}\mathrm{a}\mathrm{n}(\mathrm{X})
\bot
,
S =
\Biggl(
S1 0
S2 S3
\Biggr)
on \scrK = \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(X)
and
X =
\Biggl(
X1 0
0 0
\Biggr)
on \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(X) \rightarrow \mathrm{r}\mathrm{a}\mathrm{n}(X)\oplus \mathrm{r}\mathrm{a}\mathrm{n}(X)
\bot
.
From TX = XS, we have
T1X1 = X1S1, (2.2)
where T1 is k-quasi-M-hyponormal by Proposition 2.1, S\ast
1 is dominant by Proposition 2.4 and
X1 : \mathrm{k}\mathrm{e}\mathrm{r}(X)\bot \rightarrow \mathrm{r}\mathrm{a}\mathrm{n}(X)
is injective with dense range. First assume that T is injective. Then T1 is injective. From (2.1), S1
is injective. Since S\ast
1 is dominant, it turns out to be injective. By (2.2), we have that T \ast
1 is injective.
Ultimately, T1 is M-hyponormal. Applying Proposition 2.5 to (2.2), we obtain
T \ast
1X1 = X1S
\ast
1
and T1, S1 are normal operators. Since T1 injective, T2 = 0 by Lemma 2.1. Also S2 = 0 by
Proposition 2.4. Next assume S\ast is injective. Then S\ast
1 is injective. Then by (2.2) T \ast
1 is injective.
Ultimately, T1 turns out to be M-hyponormal. Conclude as before that
T \ast
1X1 = X1S
\ast
1
and T1, S1 are injective normal operators and so T2 = S2 = 0. Hence,
T \ast X = T \ast
1X1 = X1S
\ast
1 = XS\ast .
The rest of the proof follows from Proposition 2.2.
Corollary 2.2. Let T \in B(\scrH ) be dominant and S\ast \in B(K) be k-quasi-M-hyponormal operator.
If either T or S\ast is injective, then Fuglede – Putnam theorem holds for (T, S).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
94 S. MECHERI, T. PRASAD
Proof. From TX = XS, we have S\ast X\ast = X\ast T \ast . Applying Theorem 2.2, it follows that
SX\ast = X\ast T. The rest of the proof follows from Proposition 2.2.
Corollary 2.3. Let T \in B(\scrH ) be k-quasi-M-hyponormal operator with reducing kernel and
S\ast \in B(\scrK ) be dominant operator such that TX = XS for X \in B(\scrK ,\scrH ). Then Fuglede – Putnam
theorem holds for (T, S).
Proof. Let T \in B(\scrH ) be k-quasi-M-hyponormal with reducing kernel and S\ast \in B(\scrK ) be
dominant. We decompose T, S and X as follows:
T =
\Biggl(
T1 0
0 0
\Biggr)
on \scrH = \mathrm{k}\mathrm{e}\mathrm{r}(T )\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(T )
and
S =
\Biggl(
S1 0
0 0
\Biggr)
on \scrK = \mathrm{k}\mathrm{e}\mathrm{r}(S)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(S).
Let
X =
\Biggl(
X1 X2
X3 X4
\Biggr)
on \mathrm{k}\mathrm{e}\mathrm{r}(S)\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(S) \rightarrow \mathrm{k}\mathrm{e}\mathrm{r}(T )\bot \oplus \mathrm{k}\mathrm{e}\mathrm{r}(T ).
From TX = XS, we have \Biggl(
T1X1 T1X2
0 0
\Biggr)
=
\Biggl(
X1S1 0
X3S1 0
\Biggr)
.
The equations T1X2 = 0 and X3S1 = 0 yields X2 = X3 = 0 because T1 and S\ast
1 are injective.
Applying Theorem 2.2 to T1X1 = X1S1, it follows T \ast
1X1 = X1S
\ast
1 .
Stampfli and Wadhwa [17] proved if T be dominant and S be a normal operator and if TX = XS
where X \in B(\scrH ) has dense range, then T is a normal operator (see [17], Theorem 1). This
remarkable result for k-quasihyponormal operators has been studied by Gupta and Ramanujan [3].
Now we show this result remains true for k-quasi-M-hyponormal operators.
Theorem 2.3. Let T be a k-quasi-M-hyponormal and S a normal operator. If S is quasiaffine
transform of T, then T is a normal operator unitarily equivalent to S.
Proof. Let T be k-quasi-M-hyponormal. By Proposition 2.1, decompose T on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(T k)\oplus
\oplus \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k) as follows:
T =
\Biggl(
T1 T2
0 T3
\Biggr)
,
where T1 = T |
ran(Tk)
is M-hyponormal and T k
3 = 0. Let S1 = S|
ran(Sk)
. Decompose
S =
\Biggl(
S1 0
0 0
\Biggr)
.
Obviously, S1 is normal. Let X1 = X|
ran(Sk)
. Then
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
FUGLEDE – PUTNAM TYPE THEOREMS FOR EXTENSION OF M-HYPONORMAL OPERATORS 95
X1 : \mathrm{r}\mathrm{a}\mathrm{n}(Sk) \rightarrow \mathrm{r}\mathrm{a}\mathrm{n}(T k)
is injective and has dense range.
From TX = XS, we have
T1X1 = X1S1.
Since T1 is M-hyponormal and S1 is normal, it follows from [17] (Theorem 1) that T1 is normal
operator unitarily equivalent to S1. Consequently, \mathrm{r}\mathrm{a}\mathrm{n}(T k) reduces T and so T2 = 0 by Lemma 2.1.
Since X\ast (\mathrm{k}\mathrm{e}\mathrm{r}(T \ast k)) \subset \mathrm{k}\mathrm{e}\mathrm{r}(S\ast k) = \mathrm{k}\mathrm{e}\mathrm{r}(S\ast ),
X\ast T \ast
3 x = X\ast T \ast x = S\ast X\ast = 0
for each x \in \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k). Since X has dense range, X\ast is one-to-one. Therefore, T \ast
3 x = 0 for each
x \in \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k). Hence, T3 = 0 and so T =
\biggl(
T1 0
0 0
\biggr)
is normal.
Proposition 2.6 [16]. Let T \in B(\scrH ) be dominant and \delta \subseteq \BbbC be closed. If there exists a
bounded function f(z) : \BbbC \setminus \delta \rightarrow \scrH such that (T - zI)f(z) \equiv x for some non-zero x \in \scrH , then f(z)
is analytic on \BbbC \setminus \delta .
The above result proved for hyponormal operators by Radjabalipour [13]. This result for k-
quasihyponormal with a condition 0 /\in \delta and its consequences has been studied by Gupta [2]. In the
following theorem, we show this result hold true in the case of k-quasi-M-hyponormal operators.
Theorem 2.4. Let T \in B(\scrH ) be k-quasi-M-hyponormal and 0 /\in \delta \subseteq \BbbC be closed. If there
exists a bounded function f(\lambda ) : \BbbC \setminus \delta \rightarrow \scrH such that (T - \lambda I)f(\lambda ) \equiv x for some non-zero x \in \scrH ,
then f is analytic at every non-zero point and hence f has analytic extension everywhere on \BbbC \setminus \delta .
Proof. Suppose that T is k-quasi-M-hyponormal. By Proposition 2.1, decompose T on \scrH =
= \mathrm{r}\mathrm{a}\mathrm{n}(T k)\oplus \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k) as follows:
T =
\Biggl(
T1 T2
0 T3
\Biggr)
,
where T1 = T |
ran(Tk)
is M-hyponormal and T k
3 = 0.
Let f(\lambda ) = f1(\lambda ) \oplus f2(\lambda ) and x = x1 \oplus x2 are the decomposition of f and x, respectively.
Then
(T1 - \lambda I)f1(\lambda ) + T2f2(\lambda ) \equiv x1,
(T3 - \lambda I)f2(\lambda ) \equiv x2.
Since T k
3 = 0 and 0 /\in \delta , f2(\lambda ) = (T3 - \lambda I) - 1x2 (\lambda \not = 0) can be extended to a bounded entire func-
tion. Since k-quasi-M-hyponormal operators satisfies single valued extension property, we conclude
x2 = 0 (see [8], Proposition 1.2.16 9(f)). Hence f2(\lambda ) = 0. Therefore, for all \lambda /\in \delta ,
(T1 - \lambda I)f(\lambda ) \equiv x1.
M-hyponormality of T1 ensures f1(\lambda ) is analytic at every non zero point and has analytic extension
every where on \BbbC \setminus \delta by Proposition 2.6.
If T and T \ast are M-hyponormal, then T is normal [14]. Gupta [2] proved if T and T \ast are
k-quasihyponormal and T is injective, then T is normal. Now we establish a similar result for
k-quasi-M-hyponormal operators.
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96 S. MECHERI, T. PRASAD
Corollary 2.4. Let T be dominant or k-quasi-M-hyponormal and S\ast be k-quasi-M-hyponormal.
If either T or S is injective and S is a quasiaffine transform of T, then T and S are unitarily
equivalent normal operators. In particular, if T, T \ast are k-quasi-M-hyponormal and T is injective,
then T is normal.
Proof. Let T be dominant or k-quasi-M-hyponormal and S\ast be k-quasi-M-hyponormal. Since
S\ast is k-quasi-M-hyponormal, there exists a real positive number M such that \| (S - \lambda I)S\ast k\| \leq
\leq M\| (S - \lambda I)\ast S\ast k\| . Therefore,
Sk(S - \lambda I)\ast (S - \lambda I)S\ast k \leq M(S - \lambda I)(S - \lambda I)\ast .
Applying [14] (Theorem 2), it follows that
(S - \lambda )(S - \lambda )\ast \geq c2| (SS\ast - S\ast S)Sk| 2
for some c > 0, where | .| denote the positive part of operator. If Sk(SS\ast - S\ast S) \not = 0, then by [12]
(Theorem 1) there exists a bounded function f(\lambda ) : \BbbC \setminus \delta \rightarrow \scrH such that (S - \lambda I)f(\lambda ) \equiv x for some
non-zero x \in \scrH and so
(T - \lambda I)Xf(\lambda ) \equiv Xx.
If T is k-quasi-M-hyponormal, then, by Theorem 2.4, we have Xx = 0. If T is dominant, then we
obtain Xx = 0 by Proposition 2.6. Ultimately, x = 0, a contradiction. Therefore,
Sk(SS\ast - S\ast S) = 0.
Since S is a quasiaffine transform of T, TX = XS for injective operator X \in B(\scrH ). If T is
injective, then S is injective, Since Sk(SS\ast - S\ast S) = 0, S is normal. Then the required result
follows by Theorem 2.3.
Spectral manifold (analytic), denoted by XT (\delta ), of an operator T \in B(\scrH ) is defined as follows:
XT (\delta ) =
\bigl\{
x \in H : (T - \lambda I)f(\lambda ) \equiv x for some analytic function f(\lambda ) : \BbbC \setminus \delta \rightarrow \scrH
\bigr\}
.
If a closed subspace \scrM of \scrH is said to be hyperinvariant of T if \scrM is invariant under every operator
which commutes with T.
From Theorem 2.4, XT (\delta ) \not = \{ 0\} for k-quasi-M-hyponormal operators and it is known that
k-quasi-M-hyponormal operators satisfies single valued extension property. The above results yields
the following result by the method of [13] (Proposition 2).
Corollary 2.5. Let T \in B(\scrH ) be k-quasi-M-hyponormal and 0 /\in \delta \subseteq \BbbC be closed. If there
exists a bounded function f : \BbbC \setminus \delta \rightarrow \scrH such that (T - \lambda I)f(\lambda ) \equiv x for some non-zero x \in H,
then T has non-zero hyperinvariant subspace \scrM with \sigma (T | \scrM ) \subseteq \delta . In particular, \scrM is a nontrivial
invariant subspace of T if \delta is proper subset of \sigma (T ).
3. Quasisimilarity. Equality of spectra of quasisimilar k-quasihyponormal operators has been
proved in [3] by Gupta and Ramanujan. In Theorem 3.1, we show that spectrum of quasisimilar
k-quasi-M-hyponormal operators are same. Recall, a subspace \scrM of \scrH is called spectral maximal
space for T if \scrM contains every invariant subspace \scrC of T for which \sigma (T | \scrC ) \subset \sigma (T | \scrM ). An
operator T \in B(\scrH ) is said to be decomposable if for any finite open covering \{ U1, U1, . . . , U1\} of
spectrum of T, there exist spectral maximal subspaces \scrM 1, \scrM 2, . . . ,\scrM n of T such that
(a) \scrH = \scrM 1 +\scrM 2 + . . .+\scrM n
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FUGLEDE – PUTNAM TYPE THEOREMS FOR EXTENSION OF M-HYPONORMAL OPERATORS 97
and
(b) \sigma (T | \scrM i) \subset Ui for i = 1, 2, . . . , n.
We say that an operator T is subdecomposable operator if it is the restriction of a decomposable
operator to its invariant space (see [8]). It is well-known that T is decomposable if and only if T
has Bishop property (\beta ). The following result of Yang [20] is crucial to our purpose. It is known
that two quasisimilar M-hyponormal operators have equal spectrum.
Proposition 3.1 ([20], Corollary 2.2). Let S \in B(\scrH ) and T \in B(\scrK ) be two quasisimilar sub-
decomposible operators. Then \sigma (T ) = \sigma (S).
Theorem 3.1. If k-quasi-M-hyponormal operators T, S \in B(\scrH ) are quasisimilar, then they
have equal spectrum.
Proof. Let T, S \in B(\scrH ) be k-quasi-M-hyponormal operators. From [5], T and S satisfies
Bishop property (\beta ) and hence T and S are subdecomposible operators. Then, by Proposition 3.1,
it follows that spectrum of T and S are equal.
Two operators T \in B(\scrH ) and S \in B(\scrK ) are densely similar if there exist X \in B(\scrH ,\scrK ) and
Y \in B(\scrK ,\scrH ) such that XT = SX and Y S = TY, and are with dense ranges.
Theorem 3.2. If k-quasi-M-hyponormal operators T, S \in B(\scrH ) are densely similar, then they
have equal essential spectrum.
Proof. Since T and S are k-quasi-M-hyponormal operators, both T and S satisfies Bishop
property (\beta ). Then, by applying [8] (Theorem 3.7.13), it follows that essential spectrum of T and S
are equal.
The following result is due to Yang [20].
Proposition 3.2 ([20], Theorem 2.10). Let S \in B(\scrH ) and T \in B(\scrK ) be two quasisimilar M-
hyponormal operators. Then \sigma e(T ) = \sigma e(S).
Equality of essential spectrum of quasisimilar (p, k) quasihyponormal operators has been inves-
tigated by Kim and Kim [7]. Let MQ =
\biggl(
S Q
0 T
\biggr)
is an 2 \times 2 upper-triangular operator matrix
acting on the Hilbert space \scrH \oplus \scrK and let \sigma e(T ) denote the essential spectrum of T in B(\scrH ).
Now we prove two quasisimilar k-quasi-M-hyponormal operators have equal essential spectra.
The following result is due to Kim and Kim [7].
Proposition 3.3 [7]. Let \sigma e(S) \cap \sigma e(T ) has no interior points. Then, for every Q \in B(\scrK ,\scrH ),
\sigma e(MQ) = \sigma e(S) \cup \sigma e(T ). (3.1)
Theorem 3.3. If k-quasi-M-hyponormal operators T, S \in B(\scrH ) are quasisimilar, then they
have equal essential spectrum.
Proof. Let T, S \in B(H) be quasisimilar k-quasi-M-hyponormal operators. Then there exist
quasiaffinities X and Y such that XT = SX and Y S = TY. By Proposition 2.1, decompose T and
S as follows:
T =
\Biggl(
T1 T2
0 T3
\Biggr)
on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(T k)\oplus \mathrm{k}\mathrm{e}\mathrm{r}(T \ast k)
and
S =
\Biggl(
S1 S2
0 S3
\Biggr)
on \scrH = \mathrm{r}\mathrm{a}\mathrm{n}(Sk)\oplus \mathrm{k}\mathrm{e}\mathrm{r}(S\ast k),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
98 S. MECHERI, T. PRASAD
where T1 = T |
ran(Tk)
, S1 = T |
ran(Sk)
are M-hyponormal operators, \sigma (T ) = \sigma (T1) \cup \{ 0\} and
\sigma (S) = \sigma (S1) \cup \{ 0\} . Since quasisimilar M-hyponormal operators, have same essential spectrum
(see Proposition 3.2), in view of Propositions 2.1 and 3.3, it is enough to show that domain of T3 is
\{ 0\} if and only if domain of S3 is \{ 0\} . Since XT = SX, XT k = SkX. Let 0 \not = x \in H such that
T \ast kx = 0. Then, by the equality XT k = SkX, we have S\ast kY \ast = 0. Since Y \ast is one-to-one, we
get that domain of S3 is \{ 0\} implies domain of T3 is \{ 0\} . By a similar argument as above using the
equality Y S = TY we obtain domain of T3 is \{ 0\} implies domain of S3 is \{ 0\} .
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Received 03.02.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
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| id | umjimathkievua-article-2355 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:22Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/ca/23eb5e32bd64e40d3e4000d5d94a4dca.pdf |
| spelling | umjimathkievua-article-23552022-03-27T15:39:11Z Fuglede – Putnam type theorems for extension of $M$-hyponormal operators Fuglede – Putnam type theorems for extension of $M$-hyponormal operators Mecheri , S. Prasad , T. Mecheri , S. Prasad , T. Fuglede-Putnam theorem, normal operator, k-quasi-M-hyponormal operator, dominant operator, and quasisimilar operators. UDC 517.9 We consider $k$-quasi-$M$-hyponormal operator $T \in B(\mathcal{H})$ suchthat $TX = XS$ for some $X \in B(\mathcal{K},\mathcal{H})$ and prove the Fuglede–Putnam type theorem when adjoint of $S \in B(\mathcal{K})$ is $k$-quasi-$M$-hyponormal or dominant operators.We also show that two quasisimilar $k$-quasi-$M$-hyponormal operators have equal essential spectra. &nbsp; &nbsp; УДК 517.9Теореми типу Фугледе–Путнама для розширень $M$ -гiпонормальних операторiвРозглянуто $k$-квазі-$M$-гіпонормальний оператор $T \in B(\mathcal{H})$ такий, що $TX = XS$ для деякого $X \in B(\mathcal{K},\mathcal{H}),$ та доведено теорему типу Фугледе–Путнама, коли спряженим до $S \in B(\mathcal{K})$ є або $k$-квазі-$M$-гіпонормальний, або домінуючий оператор.Також показано, що два квазіподібні $k$-квазі-$M$-гіпонормальні оператори мають однакові суттєві спектри. Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2355 10.37863/umzh.v74i1.2355 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 89 - 98 Український математичний журнал; Том 74 № 1 (2022); 89 - 98 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2355/9178 Copyright (c) 2022 Prasad Thankarajan |
| spellingShingle | Mecheri , S. Prasad , T. Mecheri , S. Prasad , T. Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_alt | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_full | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_fullStr | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_full_unstemmed | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_short | Fuglede – Putnam type theorems for extension of $M$-hyponormal operators |
| title_sort | fuglede – putnam type theorems for extension of $m$-hyponormal operators |
| topic_facet | Fuglede-Putnam theorem normal operator k-quasi-M-hyponormal operator dominant operator and quasisimilar operators. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2355 |
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