On Linear Combination of Two Generalized Skew Projection

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1 International Mathematical Forum no On Linear Combination of Two Generalized Skew Projection S. A. Alzuraiqi Department of Mathematics Sardar Patel University Vallabh Vidyanagar Gujarat India alzoriki44@gmail.com A. B. Patel Department of Mathematics Sardar Patel University Vallabh Vidyanagar Gujarat India abp1908@yahoo.com Abstract Necessary and sufficient conditions for a linear combination of two generalized skew projections to be a generalized skew projection are developed. Keywords: normal operator projection generalized skew projection 1 Introduction Throughout this paper B(H) denotes the algebra of all bounded linear operators acting on a Hilbert space H. For given α Cand T B(H) and T B(H) symbols α T will mean the conjugate of α and the adjoint of T. T B(H) is called projection if T 2 = T = T ; we call T B(H) a generalized skew projection if T 2 = T. It is shown in [3] that for generalized skew projections S T B(H) S + T is a generalized skew projection if and only if ST = TS =0. IfS T B(H) we say that S is lower or equal to T with respect to the -order which is denoted by S T if 1. S S = S T ; 2. SS = TS. Baksalary and Baksalary [2] provided a complete solution to the problem of when a linear combination of two different idempotents is an idempotent by

2 2876 S. A. Alzuraiqi and A. B. Patel considering all situations in nonzero scalars used the linear combination. The purpose of the present paper is to consider the similar problem for generalized skew projections. 2 Main Results In the sequel we shall need following lemma. Lemma 2.1 Let S T B(H) where S is normal. Then S T if and only if SS = ST = TS. Proof. Since S T if and only if S S = S T and SS = TS. Since S is normal S T if and only if S S = S T = TS. By Putnam Fuglede s theorem [4] ST = TS and SS = ST. Hence S T if and only if SS = ST = TS. Suppose T B(H) is generalized skew projection. Then T T = T 2 T = TT. So T is normal. Also (T T ) 2 = T 2 T 2 = TT = T T. Hence T is a partial isometry and T T is an orthogonal projection onto R(T ) = R(T ) where R(T ) denotes the range of an operator T. Also T 4 = (T 2 ) 2 = (T ) 2 = ( T )=T. Theorem 2.2 Let S T B(H) be two generalized skew projections. Then T S is a generalized skew projection if and only if S T. Proof. Suppose T S is a generalized skew projection. Then (T S) 2 = (T S). Hence T 2 ST TS + S 2 = T + S. So 2S = ST + TS. Hence 2S 2 = ST +TS.Thus2S 3 = STS+S 2 T = STS+TS 2.SoS 2 T = TS 2. Since S is skew generalized projection S T = TS. By Fuglede s theorem [4] ST = TS. Hence S 2 = ST = TS. Hence by Lemma 2.1 S T. Suppose S T. Then by Lemma 2.1 S = S 2 = ST = TS. Hence 2S = ST + TS. Then by premultiply and postmultiply the last equation by S we get STS + TSS = S ST + S TS. While by premultiply and postmultiply the last equation by S we get ST = S TS + STSS SS TS TS = S STS S TS STS S respectively. Since S S = SS ST + TS = 2S TS = 2S. Hence T S is a generalized projection. Theorem 2.3 Let T 1 T 2 B(H) be distinct nonzero generalized skew projections α β be nonzero scalars and ρ = {0 1 + And let γ 1 = α α2 and γ αβ 2 = β β2. Then T = αt αβ 1 + βt 2 is a generalized skew projection if and only if any one of the following disjoint sets of conditions holds: 1. T 1 T 2 =0and α β ρ.

3 On linear combination of two generalized Skew projection T 1 T 2 = T2 and α =1 β { + } or α = 2 2 β { 3i } or α = + β { 3i T 1 T 2 =( 1 and α =1 β { 3i 1 } or α = β { + } or α = + β { 3i T 1 T 2 =( 1+ and α =1β { 3i 1+ 2 β { 3i 1 } or α = T 1 T 2 = T1 α { 3i and β =1 α { } or α = 2 β { + + i 3 } or β = } or β = + α { 3i T 1 T 2 =( and β =1 α { 3i 1 } or β = α { + } or β = + α { 3i T 1 T 2 =( and β =1α { 3i 1+ 2 α { 3i 1 } or β = } or β = 2 α { + 8. T 1 T 2 = T1 + T 2 and α β are any nonzero solutions to the equations α α 2 =2αβ = β β T 1 T 2 =( ( and α β are any nonzero solutions to the equations ( 3i)(α α 2 )=4αβ =(+ 3i)(β β 2 ). 10. T 1 T 2 =( ( and α β are any nonzero solutions to the equations (+ 3i)(α α 2 )=4αβ =( 3i)(β β 2 ). 11. T 1 T 2 = 1(γ 2 1T1 + γ 2T2 ) and α β / ρ are any nonzero solutions to the equation (γ γ 2 =( 2 2 )(γ2 2 +2γ 1) or to the equation γ γ 2 = ( 2 +i 3 2 )(γ2 2 +2γ 1) where γ γ 2 0and (consequently) γ γ T 1 T 2 T 2 T 1 and αβ(t 1 T 2 + T 2 T 1 )=(α α 2 )T1 +(β β2 )T2 where α β satisfy the equation (α α 2 )(β β 2 )=α 2 β 2. Proof. Suppose α β are nonzero complex numbers and T 1 T 2 B(H) are generalized skew projections. Then T = αt 1 + βt 2 is a generalized skew projection if and only if (αt 1 + βt 2 ) 2 = (αt 1 + βt 2 ) if and only if (αt 1 ) 2 + αβt 1 T 2 + αβt 2 T 1 +

4 2878 S. A. Alzuraiqi and A. B. Patel (βt 2 ) 2 = (αt1 + βt 2 ) if and only if α2 T1 2 + αβ(t 1T 2 + T 2 T 1 )+β 2 T2 2 = αt1 βt 2 α2 T1 + αβ(t 1T 2 + T 2 T 1 )+β 2 T2 = αt 1 βt 2 if and only if (α α 2 )T1 +(β β 2 )T2 = αβ(t 1 T 2 + T 2 T 1 ). Then the last equation is equivalent to γ 1 T1 + γ 2T2 = T 1T 2 + T 2 T 1. (1) Now premultiplying 1 by T 1 yields γ 1 T 1 T1 + γ 2T 1 T2 = T1 2T 2 + T 1 T 2 T 1 = T1 T 2 + T 1 T 2 T 1. Postmultiplying 1 by T 1 yields γ 1 T1 T 1 + γ 2 T2 T 1 = T 1 T 2 T 1 + T 2 T1 2 = T 1T 2 T 1 T 2 T1. Since T 1 T 2 are normal γ 2 (T 1 T2 T 2 T 1)=T 2 T1 T 1 T 2. Since 1 is invariant with respect to interchanging the subscripts 1 and 2 it also follows that γ 1 (T1 T 2 T 2 T1 )=T 2 T 1 T 1 T2. Hence γ 2γ 1 (T1 T 2 T 2 T1 )= γ 2 (T2 T 1 T 1 T2 )=(T 1 T 2 T 2 T1 ). Thus if 1 holds then T 1 T 2 = T 2 T 1 or γ 1γ 2 =1. (2) Now if T 1 T 2 = T 2 T 1 then by Fuglede s theorem T 1T 2 = T 2 T 1. Thus if T 2 = T and γ 1 γ 2 1 then by 1 γ 1 T 1 + γ 2 T 2 =2T 1 T 2. (3) Or T 1 T 2 T 2 T 1 and γ 1 γ 2 =1. In fact if T 1 T 2 T 2 T 1 then 1 is equivalent to that γ 1 γ 2 = 1 which is nothing but (12) of the theorem. Suppose 3 holds. Then γ 1 T1 +γ 2T2 =2T 2T 1. So that γ 1 T 2 T2 T1 + γ 2 T 2 T2 T2 =2T 2 T2 T 2 T 1. Since T 2 is a partial isometry T 2 T2 T 2 = T 2 and T2 T 2T2 = T 2 [1 p.250]. Hence γ 1T 2 T2 T 1 + γ 2T2 =2T 2T 1. Therefore γ 1 T 2 T2 T 1 +γ 2T2 =2T 2T 1 = γ 1 T1 +γ 2T2.Thusγ 1(T1 T 2T2 T 1 )=0 which is equivalent to similarly if 3 holds then we have also α α 2 =0orT 2 T 2 T 1 = T 1 (4) β β 2 =0orT 1 T 1 T 2 = T 2 (5) Since c Cis a nonzero solution to the equation c c 2 = 0 if and only if c {0 1 + } = ρ. 4 and 5 show that if T 1T 2 = T 2 T 1 then 3 is divided into the following four disjoint cases: (i) α ρ and β ρ or (ii)α ρ β / ρ and T 1 T1 T2 = T2 or (iii) α/ ρ β ρ and T 2 T2 T 1 = T 1 or (iv) α/ ρ β/ ρ T 2T2 T 1 = T 1 and T 1T1 T 2 = T 2. If (i) above holds then γ 1 = γ 2 =0. SoT 1 T 2 = 0 which is (1) of the theorem. Suppose (ii) above holds then T 1 T1 T 2 = T 2.SoT 1 T 1T 2 = T 2.Nowasα ρ γ 1 =0. So from 3 we have T2 =2γ 2 T 1 T 2.SoT 2 =2γ 2 T2 T 1. On the other hand since T 1 T 2 = T 2 T 1 T 2 = T 4 2 = (T 2 ) 2 = T 2 = (4γ 2 2 T 2 1 T 2 2 )= (4γ 2 2 T 1 T 2 ). (6)

5 On linear combination of two generalized Skew projection 2879 Thus (2γ 2 +4γ2 2 )T2 T 1 = 0. Since T 2 0 from above we can conclude that T 1 T 2 0. Hence (γ 2 +2γ2 2 ) = 0 which is equivalent to 1γ 2 2 { } we recall that 1 2 γ 2 = β β2 2αβ. (7) In 7 if we take γ 2 = 2 then from 6 we have T 1 T 2 = T2 and β β2 =2βα. Now if α = 1 then β β 2 =2β. Soβ { + If α = then β 2 2 β2 =( 3i)β. Soβ { 3i If α = + then β 2 2 β2 =(+ 3i)β. Soβ { 3i 1 This completes the proof of (2) of the theorem. Now if γ 2 =1 then T 1 T 2 =( 1 and β β 2 =(+ 3i)βα. Now if α = 1 then β { 3i 1 If α = then 2 2 β { + Ifα = + then β { 3i This completes the proof of (3) of the theorem. Similarly if γ 2 =1+ then T 2 1T 2 =( 1 + and β β 2 =( 3i)βα. Now if α = 1 then β { 3i 1 + If α = then 2 2 β { 3i 1 3 Ifα = + then β { +i This complete the proof of (4) of the theorem. Proofs (5)(6)(7) follow by replacing subscripts 1 by 2 2 by 1 and α by β β by α in (2)(3) (4) respectively. Suppose (iv) holds. On Premultiplying 3 by γ 1 T1 we get γ1t γ 1 γ 2 T1 T2 = 2γ 1 T1 T 1T 2 which is equivalent to γ1 2T 1 + γ 1 γ 2 T1 T 2 =2γ 1T 2.Soγ 1 γ 2 T1 T 2 = 2γ 1 T 2 + γ1t 2 1. Now on postmultiplying 3 by γ 2 T2 we get γ 1 γ 2 T1 T2 + γ2t = 2γ 2 T 1 T 2 T2 which is equivalent to γ 1γ 2 T1 T 2 γ2 2 T 2 =2γ 2 T 1. So γ 1 γ 2 T1 T 2 = 2γ 2 T 1 + γ2 2T 2. Hence 2γ 1 T 2 + γ1 2T 1 =2γ 2 T 1 + γ2 2T 2.So (γ 2 1 2γ 2)T 1 =(γ 2 2 2γ 1)T 2 (8). Since T 1 T 2 are assumed nonzero (γ1 2 2γ 2) = 0 if and only if (γ2 2 2γ 1)=0. In this case γ 1 γ 2 {0 2 + } (v) If γ 2 = 0 then γ 1 =0 (vi) If γ 2 = 2 then γ 1 =2 (vii) If γ 2 = + 3i then γ 1 = 3i (viii) If γ 2 = 3i then γ 1 = + 3i. Since α β / ρ γ 1 γ 2 are nonzero. Now if γ 1 = γ 2 = 2 then by 3 T 1 T 2 = T1 + T2 and α β satisfy α α2 =2αβ = β β 2 which proves (8) of the theorem. If γ 2 = + then γ 1 =. So by 3 T 1 T 2 =( ( and α β satisfy ( 3i)(α α 2 )=4αβ =( + 3i)(β β 2 ) which proves (9) of the theorem. If γ 2 = then γ 1 = +. So by 3

6 2880 S. A. Alzuraiqi and A. B. Patel T 1 T 2 =( ( + and α β satisfy (+ 3i)(α α 2 )= 4αβ =( 3i)(β β 2 ) which proves (10) of the theorem. Now if γ1 2 2γ 2 0γ2 2 2γ 1 0 let δ 12 = γ2 1 2γ 2. Then by 8 T γ2 2 2γ 2 = δ 1 12 T 1. Hence R(T 1 )=R(T 2 ) and therefore the orthogonal projection T 1 T1 = T1 3 and T 2 T2 = T 2 3 are identical. Since T 2 = δ 12 T 1 T2 3 = δ3 12 T 1 3. Since T 1 3 = T (1 δ12 3 ) = 0. Since T 1 T 2 δ Soδ 12 = or δ = + this 2 2 completes the proof of (11) of the theorem. If any of 1-12 of the theorem is satisfies then 1 holds which is equivalent to T = αt 1 + βt 2 is generalized skew projection. Suppose the scalars α β in Theorem 2.3 are real numbers. Then In (1) of the Theorem 2.3 the only choice α = β =1. In (2) the only choice is α =1β =. In (5) of the Theorem 2.3 the only choice is α = β =1. In (8) of the Theorem 2.3 the only choice is α = β = 1 3 In (12) α and β should satisfy α + β = 1 i.e. α =1 β for all real β. In other cases we do not have any choice of real α and β. ACKNOWLEDGEMENTS. S.A. Alzuraiqi is thankful to the Ministry of Higher Education and Scientific Research Republic of Yemen for scholarship to peruse his Ph.D. studies and to Thamar University Republic of Yemen for granting study leave. The authors would like to thank Professor S.J. Bhatt for helpful discussions. The UGC-SAP-DRS support to the Department of Mathematics is gratefully acknowledged by both the authors.. References [1] J. B. Conway A Course in Functional Analysis Springeg-Verlag New York [2] J.K. Baksalary O.M. Baksalary Idempotence of linear Cobinations of two idempotent matrices Linear Algebra Appl. 312 (2000) 3-7. [3] S.A. Al-Zuraiqi A.B.Patel On generalized skew projection J. of Science. 1 (2010) [4] W. Rudin Functional Analysis McGraw-Hill Received: April 2010