INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 5, NO. 2, 2015
n-Self Adjoint Operators
D. D. Barad and H. S. Mehta⋆
Abstract—The n-self adjoint operator is a generalization of
well known self-adjoint operator. We prove that any normal,
self-adjoint operator on a Hilbert space H is self-adjoint. We
also discuss the properties of Sn -the set of all n-self adjoint
operators on a Hilbert space H, which are similar to self-adjoint
operators. We obtained expected results, for some special cases
with dimH < ∞.
Index Terms—Matrices, Hilbert space, self adjoint operators
and n-self adjoint operators.
MSC 2010 Codes – 47B38, 46C05
I. I NTRODUCTION
He development of the theory of n-self adjoint operators
in infinite dimensional Hilbert space is motivated by
its connections with differential equations, particularly conjugate point theory and disconjugacy, which was first observed
by Helton [1]. The 3-self adjoint operators are studied in
([1], [2], [3], [4]).
Let H be a Hilbert space and BL(H) denote the set of
bounded, linear operators on H. An operator A ∈ BL(H) is
self adjoint if A = A∗ .
The concept of n-self adjoint operators has been defined in
[5] as follows:
Definition 1.1 An operator A ∈ BL(H) is said to be n-self
adjoint, if
n
P
(−1)k nk A∗k An−k = 0
T
k=0
0
where A and A∗0 are identity operators. ✷
We shall denote the set of all n-self adjoint operators by
Sn .
It is clear that S1 is the set of self adjoint operators.
In this paper, first we prove that if n-self adjoint operator
is normal then it becomes self-adjoint (Theorem 2.4). Then
we discuss algebraic structure of Sn . It is well known that
the set of all self-adjoint operators on H is a real, linear
subspace of BL(H) and it is closed under multiplication,
with some condition [6]. We study similar results for Sn with
dimH < ∞, in section 3.
For the basic concepts and results of matrices and operators,
we refer to [6] and [7].
II. I NFINITE D IMENSIONAL S PACE
In this section, we consider an infinite dimensional Hilbert
space H. In the main theorem, we prove that n-self adjoint
D. D. Barad, Department of Mathematics, Sardar Patel University, Vallabh
Vidyanagar 388120, India. (E-mail: [email protected])
H. S. Mehta, Department of Mathematics, Sardar Patel University, Vallabh
Vidyanagar 388120, India. (E-mail: [email protected])
⋆ The present work is partially supported by Special Assistance Programme
(SAP) of UGC, New Delhi, India.
operators together with normality give self adjointness.
We know that BL(H) is a Banach algebra with the norm
defined as
k A k= sup {k Ax k: x ∈ H, k x k= 1}, A ∈ BL(H).
First we give an example of an operator in S3 which is not
self adjoint.
Example 2.1. Define T and N on l2 as follows:
T (x) = (αx) where α = {α(i)} with α(2i) = α(2i − 1) = 1i
and
N (x) = {N (x)(i)}, where

 x(i+1) if i is odd
β(i)
N (x, t) =

0
if i is even
1
with β(i) 6= 0 ∀ i, β(i) ∈ C for some i and β(i)
→ 0 as
i → ∞.
Define A on l2 by A = T + N . Then A ∈ S3 , but A is not
self adjoint.
It is very well known that the set of self-adjoint operators
is closed in BL(H). We proved the similar result for Sn .
We fix the notation ak = (−1)k nk .
Theorem 2.2. Sn is closed in BL(H).
Proof. Let {Aj } be a sequence in Sn such that Aj → A in
BL(H). Then k Aj − A k→ 0. We shall show that A ∈ Sn .
n
P
n−k
As Aj ∈ Sn ,
ak A∗k
= 0, for each j. Further
j Aj
k=0
k
n
P
k=0
n−k
ak A∗k
−
j Aj
≤
n
P
k=0
n
P
ak A∗k An−k k
k=0
n−k
ak k A∗k
− A∗k An−k k
j Aj
Since Aj → A and adjoint operation is continuous,
A∗j → A∗ in BL(H). Also, as multiplication is jointly
n−k
continuous, A∗k
→ A∗k An−k in BL(H). So, we get
j Aj
n
n
P
P
k
ak A∗k An−k k= 0 or
ak A∗k An−k = 0. Hence
k=0
k=0
A ∈ Sn . Consequently, Sn is closed in BL(H).
To prove the main result, we shall need the following basic
result [6].
Proposition 2.3 Let A ∈ BL(H) be a normal operator.
Then, k A2 k=k A k2 . Consequently, k A2k k=k A k2k for
each k ∈ N.
Let NR denote the set of all normal operators on H.
Theorem 2.4. Sn ∩ NR = S1 .
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INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 5, NO. 2, 2015
Proof. Let A ∈ Sn ∩ NR . Then AA∗ = A∗ A and
n
P
ak A∗k An−k = 0. Since the operators A and A∗
k=0
commutes,
n
P
ak A∗k An−k = (A − A∗ )n . So (A − A∗ )n = 0.
k=0
Taking B = A − A∗ , we get B normal and also B n = 0. So
B 2n = 0. Using Proposition 2.3 for B, we get k B k= 0 or
B = 0, i. e., A = A∗ . Thus A ∈ S1 . Therefore Sn ∩ N ⊂ S1 .
It is clear that S1 ⊂ Sn ∩ NR . Consequently, Sn ∩ NR = S1 .
Note that the set Sn is not algebraically closed, even if
dimH is finite, which we shall discuss in Section 3.
The following result can be easily verified from the
definition.
Theorem 2.5. If A ∈ Sn and α ∈ R, then αA ∈ Sn .
With some extra condition, we get the following result.
Theorem 2.6. Let A ∈ Sn , B ∈ S1 and A commutes with
B. Then AB ∈ Sn .
n
P
Proof. As A ∈ Sn ,
ak A∗k An−k = 0. Also B = B ∗ and
k=0
AB = BA. So (AB)∗k = A∗k B k and
(AB)n−k = An−k B n−k = B n−k An−k . Thus
n
n
P
P
ak (AB)∗k (AB)n−k =
ak (A∗k B k )(B n−k An−k )
k=0
k=0
=[
n
P
ak A∗k An−k ]B n
k=0
=0
Theorem 3.3. Let T = αI with α ∈ R and N be an upper
(lower) triangular matrix. Then A = T + N is in Sn ,
∀n ≥ 2p − 1.
Proof. Take A = T + N . It is clear that T is self
adjoint
= (αI)N
and T N 2p−1+1
= N T . Further, if n ≥ 2p − 1,
then n+1
≥
= p. Since N is an upper (lower)
2
2
n+1
triangular matrix, we get, N [ 2 ] = 0. Therefore, by Theorem
3.1, A ∈ Sn , if n ≥ 2p − 1.
Since addition, scalar multiplication and multiplication of
upper triangular matrices are upper triangular, we get the
following result.
Theorem 3.4. Let A, B ∈ Sn be as in Theorem 3.3. Then
A + B and AB are in Sn .
Unfortunately, Sn is not closed under addition, even for
n = 3.
3 i
4 0
Example 3.5. Let A =
,B=
.
0 3
2i 4
Then A, B ∈ S3 but C = A + B 6∈ S3 , as
3
∗k 3−k
P
0
−24i
k 3
(−1) k C C
=
6= 0.
−24i
0
k=0
The next example shows that even with B = B ∗ , A + B
may not be in S3 .
2 i
1 i
Example 3.6. Let A =
,B=
.
0 2
−i 1
Then B = B ∗ and A ∈ S3 . But, A + B is not in S3 .
Hence AB ∈ Sn .
III. A LGEBRAIC P ROPERTIES
The theory of n-self adjoint operators on an infinite dimensional Hilbert space differs significantly from that of nself adjoint matrices. For example, the main characterization
(Theorem 3.1) fails for infinite dimensional space.
Throughout this section, we assume that H is a Hilbert
space with dimH = p < ∞. So, an operator A ∈ BL(H) is
considered as p × p matrix A = [aij ] and I denote the identity
operator or matrix. We discussed the algebraic properties of
Sn .
We start with the following very useful characterization of
Sn .
Theorem 3.1.[5] An operator A ∈ BL(H), with dimH <
∞, is in Sn if and only if A = T + N , where T is self adjoint,
n+1
T N = N T and N [ 2 ] = 0 ([r] denote the integer part of r).
Remarks 3.2.
1) It is clear that S2n = S2n−1 .
n+1
2) Let A ∈ Sn with A = T + N . Then, N [ 2 ] = 0 and
n+1
so, for m ≥ n, N [ 2 ] = 0. Thus A ∈ Sm , ∀m ≥ n.
3) S1 = S2 ⊂ S3 = S4 ⊂ ... ⊂ S2n−1 = S2n .
First, we show that certain matrices are always in Sn .
However, for special types of A and B in Sn , we prove
that A + B ∈ Sn .
Theorem 3.7. Let A ∈ Sn with A = T + N, B = T0 + (N )t
with T0 self adjoint and (N )t the conjugate transpose of N .
Then, A + B is self adjoint. Consequently, A + B ∈ Sn .
Proof. Let A = T + N and B = T0 + (N )t with T
n+1
and T0 self adjoint, T N = N T and N [ 2 ] = 0. Then,
A + B = (T + T0 ) + (N + (N )t ). But, for any matrix N ,
(N + (N )t ) is self adjoint. So, A + B is self adjoint. Hence
A + B ∈ Sn .
Theorem 3.8. Let A ∈ Sn . Then Ak ∈ Sn , for each k ∈ N.
Proof. Let A ∈ Sn . Then A = T + N with T = T ∗ ,
n+1
T N = N T and N [ 2 ] = 0. Now,
A2 = T 2 +2N T +N 2 = T 2 +N (2T +N ) = T 2 +N P2 (T, N ).
And, [N P2 (T, N )][
n+1
2
n+1
] = N [ n+1
2 ] P (T, N )[ 2 ] , as
2
n+1
T and N commutes. So, [N P2 (T, N )][ 2 ] = 0. Also,
T 2 [N P2 (T, N )] = [N P2 (T, N )]T 2 . Further, it is clear that T 2
is self adjoint and So, by Theorem 3.1, A2 ∈ Sn . Now A3 =
(T +N )A2 = (T +N )(T 2 +N P2 (t, N )) = T 3 +N P3 (T, N ),
where P3 (T, N ) is a polynomial in T and N . In general, for
each k, it can be checked that Ak = T k + N Pk (T, N ). So,
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INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: 2231-5330), VOL. 5, NO. 2, 2015
as we have seen earlier, T k [N Pk (T, N )] = [N Pk (T, N )]T k .
n+1
Also, T k is self adjoint and [N Pk (T, N )][ 2 ] = 0. So, by
Theorem 3.1, Ak ∈ Sn .
In general, Sn is not closed under multiplication (Example
3.6).
The next result shows that Sn is closed under adjoint.
Theorem 3.9. If A ∈ Sn , then A∗ ∈ Sn .
Proof. Let A ∈ Sn . Then A = T + N with T = T ∗ , T N =
n+1
N T and N [ 2 ] = 0. Now, A∗ = T ∗ + N ∗ = T + N ∗ . Also,
n+1
(N ∗ )[ 2 ] = 0 and T N = N T ⇒ N ∗ T = T N ∗ . Thus, by
Theorem 3.1, A∗ ∈ Sn .
IV. C ONCLUSION
We discussed the algebraic properties of the n-self adjoint
operators, Sn . We obtained two results similar to self adjoint
operators (Theorems 2.2 and 2.5). But, the sum or product
of two n-self adjoint operators may not be n-self adjoint
operator, even if dimH < ∞. However, with special form,
we get partial results (Theorems 3.3, 3.5 and 3.7). Also, we
proved that with normality, n-self adjoint operators become
self adjoint. It will be interesting to find conditions under
which Sn is closed under addition and multiplication.
R EFERENCES
[1] J. W. Helton, “Jordan operators in infinite dimensions and Sturm-Liouville
conjugate point theory”, Bulletin of the American Mathematical Society,
vol. 78, no. 1, pp. 57–61, 1972.
[2] J. W. Helton, “Infinite dimensional Jordan operators and Sturm-Liouville
conjugate point theory”, Transactions of the American Mathematical
Society, vol. 170, pp. 305–331, 1972.
[3] J. Agler, “A disconjugacy theorem for Toeplitz operators”, American
Journal of Mathematics, vol. 112, no. 1, pp. 1–14, 1990.
[4] J. Agler, “Sub-Jordan operators: Bishop’s theorem, spectral inclusion and
spectral sets”, J. Operator Theory, vol. 7, pp. 373–395, 1982.
[5] S. A. McCullough and L. Roadman, “Heriditary class of operators and
matrices”, Amer. Math. Monthly, vol. 104, no. 5, pp. 415–430, 1997.
[6] B. V. Limaye, Functional analysis, Marcel Dekkar Inc. New York, 1992.
[7] S. Lang, Introduction to Linear Algebra, Springer, New York, 1986.
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