Aluthge transform of operators on the Bergman space
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Arabian Journal of Mathematics
Chinmayee Padhy · Pabitra Kumar Jena · S. K. Paikray
Aluthge transform of operators on the Bergman space
Received: 4 June 2019 / Accepted: 29 October 2019 © The Author(s) 2019
Abstract The aim of this paper is to explore some sufficient conditions for Aluthge transform of Toeplitz operators on the Bergman space to be unitary, average of unitaries and normal. Mathematics Subject Classification
47B38 · 47B35
1 Introduction Let H , K be two separable, infinite dimensional, complex Hilbert spaces and let L(H ) denote the algebra of all bounded linear operators on H . An arbitrary operator T in L(H ) has a polar decomposition T = U |T |, where 1 |T | = (T ∗ T ) 2 and U is the appropriate partial isometry (with ker(U ) = ker(T ) and ker(U ∗ ) = ker(T ∗ )). 1 1 The Aluthge transformation (T ) = |T | 2 U |T | 2 was first introduced by Aluthge [2]. An operator T is said to be normal if T ∗ T = T T ∗ . The operator T is called quasinormal if (T T ∗ )T = (T ∗ T )T . An operator T is called hyponormal if T ∗ T ≥ T T ∗ . An operator T is paranormal if T x2 ≤ T 2 xx. Here, we denote ∗ (T ) = ((T ))∗ for T ∈ L(H ). Let sp(T ) denotes the spectrum of T . Define m(T ) = inf{T x : x = 1}. An operator T is called w-hyponormal if |(T )| ≥ |T | ≥ |∗ (T )|. An operator T is said to be binormal if [|T |, |T ∗ |] = 0, where [T, S] = T S − ST. The tensor product of x ∈ H and y ∈ K is a conjugate bilinear functional x ⊗ y : H × K −→ C defined by (x ⊗ y)(u, v) = x, u y, v for every (u, v) ∈ H × K . The collection of all (finite) sums of tensors xi ⊗ yi with xi ∈ H and yi ∈ K , denoted by H ⊗ K , is a complex linear N space equipped with an inner product ., . : (H ⊗ K ) × (H ⊗ K ) −→ C defined for arbitrary i=1 xi ⊗ yi , N M M M N (the same and j=1 w j ⊗ z j in H ⊗ K , by i=1 x i ⊗ yi , j=1 w j ⊗ z j = i=1 j=1 x i , w j yi , z j notation for the inner products on H, K and H ⊗ K ). The tensor product on H ⊗ K of two operators T in L(H ) N N xi ⊗ yi = i=1 T xi ⊗ Syi and S in L(K ) is the operator T ⊗ S : H ⊗ K −→ H ⊗ K defined by (T ⊗ S) i=1 N for every i=1 xi ⊗ yi ∈ H ⊗ K , which lies in L(H ⊗ K ). The complete inner product space H ⊗ K is ˆ , which is the tensor product space of H and K . The extension of T ⊗ S over the Hilbert space denoted by H ⊗K C. Padhy · S. K. Paikray Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha 768018, India E-mail: [email protected]
S. K. Paikray E-mail: [email protected] P. K. Jena (B) P. G. Department of Mathematics, Berhampur University, Bhanjabihar, Berhampur, Ganjam, Odisha 760007, India E-mail: [email protected]
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ˆ denoted by T ⊗S ˆ is the tensor product of T and S on the tensor product space, which lies in L(H ⊗K ˆ ) H ⊗K (see [6], [12] and [17]). Definition 1.1 If the sequence {· · · , T 3 (T 3 )∗ , T 2 (T 2 )∗ , T T ∗ , T ∗ T, (T 2 )∗ T 2 , (T 3 )∗ T 3 , (T 4 )∗ T 4 , · · · } is commutative, the operator T ∈ L(H ) is called centered operator (see [10]). D
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