Skew normal operators acting on a Hilbert space

The motive of the present study is to analyze the basic properties of skew normal operators acting on a Hilbert space H. New theorems and examples are given on some properties in the Hilbert space using skew normal operators.

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Introduction

Let H be a Hilbert space and L (H) be the algebra of all bounded linear operators acting on H. The basic properties of various operators have been generalized by different authors. The classes of normal and skew normal operators have received greater importance in the study of operator theory. Several interesting aspects regarding these operators are investigated by many outstanding researchers [1, 3, 6].

A skew normal operator is a generalization of normal operator. In this paper such kind of operators are examined and results are derived. 2. Skew Normal Operator An operator ) ( H L T  is called self-adjointif T T  , normal if   TT T T ,i.e., T commutes with T [2], quasinormal if     T T T T T T    [5]. The class of Quasinormal operators was first introduced and studied by A.Brown.

Panayappan and Sivamani [2] discussed about n-binormal operators. Meenambika and Seshaiah [3, 4] have worked on Aluthge transformation of Semi-hyponormaloperators. An operator T is called isometry if I T T   ,    2-isometry if 0 2 2 2   I T T T T . An operator ) ( H L T  is skew normal if     T T T T TT    i.e., T commutes with normal operator and it is denoted by [sN].

The proposed method analyses some of the relationship between self-adjoint and skew-normal operators, also some results are proved based on skew-normal operator. Theorem: 2.1 If   T sN  then so are i) kT for any real number k ii) The restriction T/M of T to any closed subspace M of H that reduces T. Proof: i) The proof is obvious. ii) If T is skew normal then     T T T T TT    Consider,   M T M TT M T M T M T / / / / /             M T TT / Maths 2018; 3(5): 128-133 K Meenambika Department of Mathematics, Sri Shanmugha College of Engineering and Technology, Tamilnadu, India. Dr.

CV Seshaiah Department of Mathematics, GMR Institute of Technology, GMR Nagar, Rajam, Andhra Pradesh, India Dr. N Sivamani Department of Mathematics, Tamilnadu College of Engineering, Coimbatore, Tamil Nadu, India Correspondence K Meenambika Department of Mathematics, Sri Shanmugha College of Engineering and Technology, Tamilnadu, India.   / / / M T M T M T   / / / M T M T M T      / / /       M T M T M T Hence   / T M sN  Theorem: 2.2 Let ) ( H L T  be skew normal operators which is unitarily equivalent to an operator S if and only if TU=UT, T U TU   ,   UT U T .Then S is skew normal. Proof: Since T is unitarily equivalent to S,there is a unitary operator U such that TU U S   which implies U T U S    .We must    .Since T is skew normal then     T T T T TT show that     S S S S SS    Now,   S S S S SS              TU UU T U TU U TU U U T TUU U            TU T U TU U TU U U TT U            TU UT By TU T U UT U UT U U TT U   ) (         TU T U T T U TT U             UT U T T U TU By UT U T T T UT TU    ) , (            T T T T TT Therefore S is Skew Normal. Theorem: 2.3 If ) ( H L T  is normal then,it is skew-normal.

Proof: An operator T is normal if   TT T T When T commutes with normal operator, it becomes,     T T T T TT    Hence T is skew normal. The following example shows that the converse need not be true. Example: 2.4   2 1 T is skew normal but not normal operator. 1 0 Let      Theorem: 2.5 Let     T sN and S sN   . If T and S are doubly commuting then TS is skew-normal.

Proof: Consider,    TS T TSS TS TS TS             TS S STT      TS S STT        TS T S S TS STT      S TS TT S ) (     ) (   Normal Skew is T S S T T ST    S TS STT        T S TS TS ST TS S TST  ,            S T T S TS T TSS        TS ST ST T TSS          S T ST T ST TSS      T ST SS T       Normal Skew is S T T S S TS        T ST TSS        S T ST ST T S TS              TS TS TS Hence, TS is skew-normal. Example: 2.6    1 1 T be not commuting skew normal operators. Then ST is not skew normal.   1 0 0 1 Let   0 1 S and         The following example shows that the sum and difference of two commuting skew normal operators need not be skew normal. Example: 2.7        31 12 T S is 1 0 2 1 27 17       Let   2 1 S and   0 0 T on . 2 R Then S and T are commuting skew normal operators but       13 8 T S is not skew normal.     8 5    not skew normal and   Theorem: 2.8 Let T U T  be the polar decomposition of an operator T.

Then T U T  is skew normal if U T T U  Proof: Assume U T T U  , then     T U U T T U T U U T T U T T T T TT            2 2    T T U T U U T U     2 2    T T U T U UU T     2 2   T T U T U T      0 So T is skew normal. Theorem: 2.9 2 2 A B AB i )  Let ) ( H L T  with the Cartesian decomposition T=A+iB, then T is skew-normal if and only if 2 2 BA B A ii )  Proof: Since T is skew-normal,     T T T T TT    Consider, iB A iB A iB A T TT              2 2    iB A iB A      2 2    iB A B A    3 2 2 2     iB B iA A B A A     iB iBA AB A ) 1 ...( .......... .......... 3 2 2 3 iB A iB A iB A TT T              2 2    iB A iB A       2 2    B A iB A    2 2 2 2     iBB iBA AB AA     iB iBA AB A ) 2 ...( .......... .......... 3 2 2 3 From (1) and (2), T is Skew Normal operator. The following example shows that skew normal operator need not be isometric operator. Example: 2.10   2 1 T which is a skew-normal but not isometric operator. 1 0    Consider the operator   3.

Self-adjoint and Skew-normal operators Theorem: 3.1 If T is skew normal and is any scalar which is real then T  is also skew-normal operator. Proof: Since T is skew normal operator,we have,        T T T T TT If is any scalar which is real,then     T T   Consider,      3 ...................(3) T T T T T T              3     T T T T T T              3 .................(4 ) (S in ) T T T ce T is skew n o rm a l   From (3) and (4), T is skew –normal. Theorem: 3.2 If T is skew-normal operator which is a self-adjoint operator, then  T is also skew-normal operator. Proof: Since T is skew normal operator, we have,     T T T T TT    …………………………………. (5) Since T is self-adjoint we have T T   Replace T by T  in (5), Consider,      T T T T T T     TTT ) 6 .......( .......... ..........      T TT T T T     TT T   TTT ) 7 .......( .......... ..........

From (6) and (7),  T self adjoint operator. Theorem: 3.3 If T is self adjointoperator, then T is skew-normal operator. Proof: Since T is self adjoint operator, T T   Now,  3 T T TT T TT   ) 8 .......( ..........      3 T TT T T T T       ) 9 .......( .......... From (8) and (9),T is skew normal operator.

Theorem: 3.4 Let T be any operator on a Hilbert space H.Then i)    T T is skew normal. ii)  TT is skew normal. iii) T T  is skew normal. iv)     TT I T T I , are skew normal. Proof: i. Let    T T N       N T T T T N      Hence, N is a self adjoint operator. We know that every self adjoint operator is skew-normal.

Therefore,    T T N is skew –normal operator. ii.           TT T T TT a. Hence,  TT is self-adjoint operator, so  TT is skew-normal operator. iii.   T T T T T T         a. Hence, T T  is self-adjoint operator, so T T  is skew-normal operator. iv.       T T I T T I T T I                               TT I T T I TT I Hence,     TT I T T I , are self-adjoint operator,so     TT I T T I , areskew-normal operators. Theorem: 3.5 If T is a self -adjoint operator, then 1  T is also a skew normal operator.

Proof: Since T is self adjoint operator, T T   To prove 1  T is a skew normal operator, it is enough to check if it is a self -adjoint operator. Now, 1 1 1        T T T     1  T Self-adjoint operator. Hence, 1  T is skew-normal operator. Proof verification Assume that 1  T is self-adjoint operator. 1 1      T T   To prove that 1  T is skew-normal operator, Consider,        T T T T T T     1 1 1 1 1 1            1 3  ( ) T  3  T .......................(1 0 )        T T T T T T     1 1 1 1 1 1             1 3  ( ) T  3  T .......................(1 1) From (10) and (11), 1  T is skew-normal operator.

Theorem: 3.6 Let T be a self-adjoint operator on a Hilbert space H,then TS S  is skew normal. Proof: Since T is self adjoint operator, T T   Consider,            TS S S T S TS S TS S   is self adjoint. By theorem (2.3), we have the result that if TS S  is self-adjoint, then it is skew-normal. Proof verification Assume TS S  is self adjoint, Consider,        S T S S T S S T S S T S S T S S T S                    3   S T S ....................................(1 2 )          S T S S T S S T S S T S S T S S T S                    3   S T S ....................................(1 3)   From (12) and (13), TS S  is skew normal operator.

Conclusion

This paper has presented a new class of operators called Skew normal operators. Some of the characters of skew normal operators were studied. The described work is focused on relationship between self adjoint and skew normal operators. As for future work is concerned, this class of operators is extended to higher class of operators.

1. Laith Shaakir K, Elaf Abdulwahid S. Skew N-Normal operators, Aust. J Basic & Appl. Sci, 2014, 340-44.
2. Panayappan S, Sivamani N. On n-binormal Operators, General Mathematics Notes, 2012, 1-8.
3. Seshaiah CV, Meenambika K. A generalization of Aluthge transformation using semi-hyponormal operators. International Journal for Research in Mathematics and Statistics. 2016, 18-28.
4. Seshaiah CV, Meenambika K. Factorization of semi-hyponormal operators. Journal of faculty of agronomy. 2018; 34(2):306- 09.
5. Veluchamy T, Manikandan KM. N-Power quasi normal operators on the Hilbert space. IOSR Journal of Mathematics. 2016, 06-09.
6. Berberian SK. Introduction to Hilbert Space (Chelsea Publ. Co., New York), 1976.