On generalized Davis-Wielandt radius inequalities of semi-Hilbertian space operators
aa r X i v : . [ m a t h . F A ] J un ON GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIESOF SEMI-HILBERTIAN SPACE OPERATORS
ANIKET BHANJA, PINTU BHUNIA AND KALLOL PAUL
Abstract.
Let A be a positive (semidefinite) operator on a complex Hilbertspace H and let A = (cid:18) A OO A (cid:19) . We obtain upper and lower bounds forthe A -Davis-Wielandt radius of semi-Hilbertian space operators, which gen-eralize and improve on the existing ones. We also obtain upper bounds forthe A -Davis-Wielandt radius of 2 × A -Davis-Wielandt radius of two operator matrices (cid:18) I X (cid:19) and (cid:18) X (cid:19) , where X is a semi-Hilbertian space operator. Introduction and Preliminaries
Let B ( H ) denote the C ∗ -algebra of all bounded linear operators acting on a complexHilbert space H with usual inner product h· , ·i and the corresponding norm k · k .The letters I and O stand for the identity operator and the zero operator on H ,respectively. For T ∈ B ( H ), we denote by R ( T ) and N ( T ) the range and the nullspace of T , respectively. By R ( T ) we denote the norm closure of R ( T ). Let T ∗ bethe adjoint of T . The cone of all positive (semidefinite) operators is given by: B ( H ) + = { A ∈ B ( H ) : h Ax, x i ≥ , ∀ x ∈ H } . Every A ∈ B ( H ) + defines the following positive semidefinite sesquilinear form: h· , ·i A : H × H −→ C , ( x, y ) x, y i A = h Ax, y i , and the seminorm induced by the above sesquilinear form is given by: k x k A = p h x, x i A , x ∈ H . This makes H into a semi-Hilbertian space. It is easy to observe that k x k A = 0 ifand only if x ∈ N ( A ). Therefore, k · k A is a norm on H if and only if A is injective.Also we observe that ( H , k · k A ) is complete if and only if R ( A ) is closed in H .Let us fix the alphabet A for positive (semidefinite) operator on H and we also fix A = (cid:18) A OO A (cid:19) . Definition 1.1.
Let T ∈ B ( H ). An operator S ∈ B ( H ) is called an A -adjoint of T if the equality h T x, y i A = h x, Sy i A holds, for all x, y ∈ H . Mathematics Subject Classification.
Primary 47A12, 46C05, Secondary 47A30, 47A50.
Key words and phrases. A -Davis-Wielandt radius, A -numerical radius, A -operator seminorm,Semi-Hilbertian space.Pintu Bhunia would like to thank UGC, Govt. of India for the financial support in the formof SRF. Prof. Paul would like to thank RUSA 2.0, Jadavpur University for partial support. Therefore, S is an A -adjoint of T if and only if S is a solution of the equation AX = T ∗ A in B ( H ). For T ∈ B ( H ), the existence of an A -adjoint of T is notguaranteed. The set of all operators acting on H that admit A -adjoints is denotedby B A ( H ). It follows from Douglas Theorem [8] that B A ( H ) = { T ∈ B ( H ) : R ( T ∗ A ) ⊆ R ( A ) } . By Douglas Theorem [8], we have if T ∈ B A ( H ) then the operator equation AX = T ∗ A has a unique solution, denoted by T ♯ A , satisfying R ( T ♯ A ) ⊆ R ( A ). Note that T ♯ A = A † T ∗ A , where A † is the Moore-Penrose inverse of A (see [2]). Also, we have AT ♯ A = T ∗ A. An operator T ∈ B ( H ) is said to be A -bounded if there exists c > k T x k A ≤ c k x k A , for all x ∈ H . We observe that B A / ( H ) is the collectionof all A -bounded operators, i.e., B A / ( H ) = { T ∈ B ( H ) : ∃ c > k T x k A ≤ c k x k A , ∀ x ∈ H} . It is well-known that B A ( H ) and B A / ( H ) are two subalgebras of B ( H ) which areneither closed nor dense in B ( H ). Moreover, the following inclusions B A ( H ) ⊆ B A / ( H ) ⊆ B ( H )hold with equality if A is injective and has closed range. Let us now define A -selfadjoint, A -normal and A -unitary operators. Definition 1.2.
An operator T ∈ B ( H ) is called A -selfadjoint if AT is selfadjoint,i.e., AT = T ∗ A and it is called A -positive if AT ≥ T is A -selfadjoint then T ∈ B A ( H ). However, in general, it doesnot always imply T = T ♯ A . An operator T ∈ B A ( H ) satisfies T = T ♯ A if and onlyif T is A -selfadjoint and R ( T ) ⊆ R ( A ). Definition 1.3.
An operator T ∈ B A ( H ) is said to be A -normal if T T ♯ A = T ♯ A T .We know that every selfadjoint operator is normal. But, an A -selfadjoint operatoris not necessarily A -normal (see [3, Example 5.1]). Definition 1.4.
An operator U ∈ B A ( H ) is said to be A -unitary if k U x k A = k U ♯ A x k A = k x k A , for all x ∈ H .It was shown in [1] that an operator U ∈ B A ( H ) is A -unitary if and only if U ♯ A U =( U ♯ A ) ♯ A U ♯ A = P A , where P A denotes the projection onto R ( A ). We mention herethat if T ∈ B A ( H ) then T ♯ A ∈ B A ( H ) and ( T ♯ A ) ♯ A = P A T P A .Let T ∈ B A / ( H ). The A -operator seminorm and the A -minimum modulus of T are defined respectively as: k T k A = sup (cid:26) k T x k A k x k A : x ∈R ( A ) , x =0 (cid:27) = sup {k T x k A : x ∈ H , k x k A = 1 } ,m A ( T ) = inf (cid:26) k T x k A k x k A : x ∈R ( A ) , x =0 (cid:27) = inf {k T x k A : x ∈ H , k x k A = 1 } . Let T ∈ B A / ( H ). The A -numerical range, the A -numerical radius and the A -Crawford number of T are defined respectively as: W A ( T ) = {h T x, x i A : x ∈ H , k x k A = 1 } ,w A ( T ) = sup {| c | : c ∈ W A ( T ) } and c A ( T ) = inf {| c | : c ∈ W A ( T ) } . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 3
The A -operator seminorm attainment set of T, denoted as M AT , is defined as theset of all A -unit vectors in H at which T attains its A -operator seminorm, i.e., M AT = { x ∈ H : k T x k A = k T k A , k x k A = 1 } . Likewise the A -numerical radius attainment set and the A -Crawford number at-tainment set of T, denoted as W AT and c AT respectively, are defined as W AT = { x ∈ H : |h T x, x i A | = w A ( T ) , k x k A = 1 } ,c AT = { x ∈ H : |h T x, x i A | = c A ( T ) , k x k A = 1 } . It is well known that k · k A and w A ( · ) are equivalent seminorm on B A / ( H ), satis-fying the following inequality: k T k A ≤ w A ( T ) ≤ k T k A , T ∈ B A / ( H ) . The first inequality becomes equality if AT = O and the second inequality becomesequality if T is A -normal (see [9]). For T ∈ B A ( H ), we write Re A ( T ) = ( T + T ♯ A )and Im A ( T ) = ( T − T ♯ A ). For every A -selfadjoint operator T , we have (see [15]) w A ( T ) = k T k A . Also T ♯ A T , T T ♯ A are A-selfadjoint and A-positive operators satisfying the followingequality: k T ♯ A T k A = k T T ♯ A k A = k T k A = k T ♯ A k A . For T ∈ B A ( H ), we write | T | A = T ♯ A T . For T, S ∈ B A ( H ), ( T S ) ♯ A = S ♯ A T ♯ A , k T S k A ≤ k T k A k S k A and k T x k A ≤ k T k A k x k A , for all x ∈ H . For further readingswe refer the readers to [1, 2].Motivated by the study of the A -numerical radius of semi-Hilbertian space opera-tors, we here study the A -Davis-Wielandt radius of semi-Hilbertian space operators.This is a generalization of the Davis-Wielandt radius of Hilbert space operators.The Davis-Wielandt shell and the Davis-Wielandt radius of an operator T ∈ B ( H )are defined respectively as (see [7, 18]): DW ( T ) = (cid:8)(cid:0) h T x, x i , k T x k (cid:1) : x ∈ H , k x k = 1 (cid:9) ,dw ( T ) = sup np |h T x, x i| + k T x k : x ∈ H , k x k = 1 o . Recently many mathematicians [12, 13, 14, 16, 17] have studied the Davis-Wielandtshell and the Davis-Wielandt radius of an operator T ∈ B ( H ). The A -Davis-Wielandt shell and the A -Davis-Wielandt radius of an operator T ∈ B A / ( H ) aredefined respectively as (see [11]): DW A ( T ) = (cid:8)(cid:0) h T x, x i A , k T x k A (cid:1) : x ∈ H , k x k A = 1 (cid:9) ,dw A ( T ) = sup (cid:26)q |h T x, x i A | + k T x k A : x ∈ H , k x k A = 1 (cid:27) . It is easy to see that the A -Davis-Wielandt radius of T ∈ B A / ( H ) satisfying thefollowing inequality:max { w A ( T ) , k T k A } ≤ dw A ( T ) ≤ q w A ( T ) + k T k A . (1)Recently, Feki in [10] have obtained some upper bounds for the A -Davis-Wielandtradius of operators in B A ( H ) . In this paper, we study about the equality of thelower bounds for the A -Davis-Wielandt radius of A -bounded operators mentioned ANIKET BHANJA, PINTU BHUNIA AND KALLOL PAUL in (1). We obtain upper and lower bounds for the A -Davis-Wielandt radius ofoperators in B A ( H ) , which generalize and improve on the existing ones. Further weobtain inequalities for the A -Davis-Wielandt radius of 2 × B A ( H ⊕ H ) , which generalize inequalities in [4]. Next, we obtain an upper boundfor the A -Davis-Wielandt radius of sum of product operators in B A ( H ) , i.e., if P, Q, X, Y ∈ B A ( H ) then for any t ∈ R \ { } , we have dw A ( P XQ ♯ A ± QY P ♯ A ) ≤ ( t k P k A + 1 t k Q k A ) { ( t k P X k A + 1 t k QY k A ) + α } , where α = w A (cid:18) O XY O (cid:19) . Finally, we compute the exact value for the A -Davis-Wielandt radius of two operator matrices (cid:18) I X (cid:19) and (cid:18) X (cid:19) , where X ∈B A / ( H ). 2. Main results
We begin this section with the study of the equality conditions of both upper andlower bounds of A -bounded operators mentioned in (1). Fisrt we mention thefollowing known result (see [11, Th. 11 and Prop. 4]). Theorem 2.1.
Let T ∈ B A / ( H ) . Then the following conditions are equivalent: ( i ) dw A ( T ) = p w A ( T ) + k T k A . ( ii ) T is A -normaloid, i.e, w A ( T ) = k T k A . ( iii ) There exist a sequence of A -unit vectors { x n } in H such that lim n →∞ k T x n k A = k T k A and lim n →∞ |h T x n , x n i A | = w A ( T ) . Remark 2.2. If H is finite-dimensional then condition (iii) of Theorem 2.1 isreplaced by M AT ∩ W AT = ∅ , i.e., there exist a A -unit vector x in H such that k T x k A = k T k A and |h T x, x i A | = w A ( T ) . Next we find the equality condition of the first inequality in (1).
Theorem 2.3.
Let T ∈ B A / ( H ) . Then the following conditions are equivalent: ( i ) dw A ( T ) = w A ( T ) . ( ii ) AT = 0 .Proof. The part ( ii ) ⇒ ( i ) follows trivially. We only prove ( i ) ⇒ ( ii ) . Since T ∈ B A / ( H ) , there exists a sequence { x n } in H with k x n k A = 1 such that w A ( T ) = lim n →∞ |h T x n , x n i A | . The sequence {k T x n k A } , being a bounded sequenceof real numbers has a convergent subsequence {k T x n k k A } . Now w A ( T ) = dw A ( T ) ≥|h T x n k , x n k i A | + k T x n k k A . Taking limit on both sides, we get w A ( T ) = dw A ( T ) ≥ w A ( T ) + lim k →∞ k T x n k k A . This implies that lim k →∞ k T x n k k A = 0. Therefore, itfollows from Cauchy-Schwarz inequality that w A ( T ) = lim k →∞ |h T x n k , x n k i A | ≤ lim k →∞ k T x n k k A = 0 . So, we get w A ( T ) = 0 and hence, AT = 0 . (cid:3) Theorem 2.4.
Let T ∈ B A / ( H ) and dw A ( T ) = k T k A . Then either of the follow-ing condition holds: ( i ) Let M AT = ∅ . Then |h T x, x i A | = 0 if x ∈ M AT , i.e., M AT ⊆ c AT . ( ii ) Let M AT = ∅ . Then there exists a sequence { x n } in H with k x n k A = 1 suchthat lim n →∞ k T x n k A = k T k A and lim n →∞ |h T x n , x n i A | = 0 . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 5
Proof. ( i ) Let M AT = ∅ and x ∈ M AT . So, k T x k A = k T k A = dw A ( T ) ≥ |h T x, x i A | + k T x k A . This implies that |h T x, x i A | = 0 . so x ∈ c AT . Therefore, M AT ⊆ c AT . ( ii ) Let M AT = ∅ . Since T ∈ B A / ( H ) , there exists a sequence { x n } in H with k x n k A = 1 such that k T k A = lim n →∞ k T x n k A . Since {|h
T x n , x n i A |} is a boundedsequence of scalars, so it has a convergent subsequence {|h T x n k , x n k i A |} . Now k T k A = dw A ( T ) ≥ |h T x n k , x n k i A | + k T x n k k A . Taking limit on both sides, we get k T k A = dw A ( T ) ≥ lim k →∞ |h T x n k , x n k i A | + k T k A and so, lim k →∞ |h T x n k , x n k i A | =0 . This completes the proof. (cid:3)
Remark 2.5.
We note that the converse part of Theorem 2.4 may not hold, (see[4, Remark 2.3].We next obtain lower bounds for the A -Davis-Wielandt radius of operators in B A ( H ). Theorem 2.6.
Let T ∈ B A ( H ) . Then ( i ) dw A ( T ) ≥ max (cid:8) w A ( T ) + c A ( | T | A ) , k T k A + c A ( T ) (cid:9) , ( ii ) dw A ( T ) ≥ (cid:8) w A ( T ) c A ( | T | A ) , c A ( T ) k T k A (cid:9) . Proof. ( i ) Let x be a A -unit vector in H . Then from the definition of dw A ( T ), weget dw A ( T ) ≥ |h T x, x i A | + k T x k A = |h T x, x i A | + h| T | A x, x i A ≥ |h T x, x i A | + c A ( | T | A ) . Therefore, taking supremum over all A -unit vectors in H , we have dw A ( T ) ≥ w A ( T ) + c A ( | T | A ) . Again from dw A ( T ) ≥ |h T x, x i A | + k T x k A , where k x k A = 1, we get dw A ( T ) ≥ c A ( T ) + k T x k A . Taking supremum over all A -unit vectors in H , we have dw A ( T ) ≥ c A ( T ) + k T k A . This completes the proof of ( i ) . ( ii ) For all x ∈ H with k x k A = 1 , we have |h T x, x i A | + k T x k A ≥ |h T x, x i A |k T x k A and so, dw A ( T ) ≥ |h T x, x i A |h| T | A x, x i A ≥ |h T x, x i A | c A ( | T | A ) . Taking supremum over all A -unit vectors in H , we get dw A ( T ) ≥ w A ( T ) c A ( | T | A ) . Again from |h T x, x i A | + k T x k A ≥ |h T x, x i A |k T x k A , we have dw A ( T ) ≥ c A ( T ) k T x k A . Taking supremum over all A -unit vectors in H , we get dw A ( T ) ≥ c A ( T ) k T k A . This completes the proof. (cid:3)
ANIKET BHANJA, PINTU BHUNIA AND KALLOL PAUL
Remark 2.7. (i) It is easy to observe that the lower bound of the A -Davis-Wielandtradius of T ∈ B A ( H ) obtained in Theorem 2.6 (i) is sharper than that in (1).(ii) Also, both the inequalities in [4, Th. 2.4] follow from Theorem 2.6 by considering A = I .In the following theorem we obtain an upper bound for the A -Davis-Wielandt radiusof operators in B A ( H ). Theorem 2.8.
Let T ∈ B A ( H ) . Then dw A ( T ) ≤ sup θ ∈ R w A ( e i θ T + | T | A ) − c A ( T ) m A ( T ) . Proof.
Let x ∈ H with k x k A = 1 . Then there exists θ ∈ R such that |h T x, x i A | = e i θ h T x, x i A . Now, |h T x, x i A | + k T x k A = h e i θ T x, x i A + h| T | A x, x i A = ( h e i θ T x, x i A + h| T | A x, x i A ) − h e i θ T x, x i A h| T | A x, x i A . Hence,2 h e i θ T x, x i A h| T | A x, x i A + |h T x, x i A | + k T x k A = (cid:0) h e i θ T x, x i A + h| T | A x, x i A (cid:1) ⇒ h e i θ T x, x i A h| T | A x, x i A + |h T x, x i A | + k T x k A = h ( e i θ T + | T | A ) x, x i A ⇒ |h T x, x i A |h| T | A x, x i A + |h T x, x i A | + k T x k A ≤ w A ( e i θ T + | T | A ) . Therefore,2 |h T x, x i A | h| T | A x, x i A + |h T x, x i A | + k T x k A ≤ sup θ ∈ R w A ( e i θ T + | T | A )and so, 2 c A ( T ) m A ( T ) + |h T x, x i A | + k T x k A ≤ sup θ ∈ R w A ( e i θ T + | T | A ) . Hence, taking supremum over all A -unit vectors in H , we get2 c A ( T ) m A ( T ) + dw A ( T ) ≤ sup θ ∈ R w A ( e i θ T + | T | A ) . ⇒ dw A ( T ) ≤ sup θ ∈ R w A ( e i θ T + | T | A ) − c A ( T ) m A ( T ) . (cid:3) Remark 2.9.
We would like to note that the inequality in [4, Th. 2.6] follows fromTheorem 2.8 by considering A = I .Next we obtain the following upper and lower bounds for the A -Davis-Wielandtradius of operators in B A ( H ). Theorem 2.10.
Let T ∈ B A ( H ) . Then (cid:8) w A ( T + | T | A ) + c A ( T − | T | A ) (cid:9) ≤ dw A ( T ) ≤ (cid:8) w A ( T + | T | A ) + w A ( T − | T | A ) (cid:9) . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 7
Proof.
Let x ∈ H with k x k A = 1 . Then |h T x, x i A | + k T x k A = 12 |h T x, x i A + h T x, T x i A | + 12 |h T x, x i A − h T x, T x i A | = 12 (cid:12)(cid:12) h T x, x i A + h| T | A x, x i A (cid:12)(cid:12) + 12 (cid:12)(cid:12) h T x, x i A − h| T | A x, x i A (cid:12)(cid:12) = 12 (cid:12)(cid:12) h ( T + | T | A ) x, x i A (cid:12)(cid:12) + 12 (cid:12)(cid:12) h ( T − | T | A ) x, x i A (cid:12)(cid:12) ≥ n(cid:12)(cid:12) h ( T + | T | A ) x, x i A (cid:12)(cid:12) + c A ( T − | T | A ) o . Therefore, taking supremum over all A -unit vectors in H , we get dw A ( T ) ≥ (cid:8) w A ( T + | T | A ) + c A ( T − | T | A ) (cid:9) . Again, |h T x, x i A | + k T x k A = 12 |h T x, x i A + h T x, T x i A | + 12 |h T x, x i A − h T x, T x i A | = 12 (cid:12)(cid:12) h T x, x i A + h| T | A x, x i A (cid:12)(cid:12) + 12 (cid:12)(cid:12) h T x, x i A − h| T | A x, x i A (cid:12)(cid:12) = 12 (cid:12)(cid:12) h ( T + | T | A ) x, x i A (cid:12)(cid:12) + 12 (cid:12)(cid:12) h ( T − | T | A ) x, x i A (cid:12)(cid:12) ≤ (cid:8) w A ( T + | T | A ) + w A ( T − | T | A ) (cid:9) . Therefore, taking supremum over all A -unit vectors in H , we get dw A ( T ) ≤ (cid:8) w A ( T + | T | A ) + w A ( T − | T | A ) (cid:9) . Hence completes the proof. (cid:3)
Remark 2.11.
We would like to remark that the inequality obtained in Theorem2.10 is generalizes the inequality in [4, Th. 2.8].In the next theorem we obtain upper bounds for the A -Davis-Wielandt radius of T ∈ B A ( H ). First we need the following lemma. Lemma 2.12.
Let x, y, e ∈ H with k e k A = 1 . Then |h x, e i A h e, y i A | ≤
12 ( |h x, y i A | + k x k A k y k A ) . Proof.
For all a, b, c, d ∈ R , we have ( ac − bd ) ≥ ( a − b )( c − d ). Using this andthe Cauchy Schwarz inequality, we get | h x − h x, e i A e, y − h y, e i A e i A | ≤ k x − h x, e i A e k A k y − h y, e i A e k A = ⇒ |h x, y i A − h x, e i A h e, y i A | ≤ ( k x k A − |h x, e i A | )( k y k A − |h y, e i A | )= ⇒ |h x, y i A − h x, e i A h e, y i A | ≤ ( k x k A k y k A − |h x, e i A ||h y, e i A | ) . Since |h x, e i A | ≤ k x k A and |h y, e i A | ≤ k y k A , so ( k x k A k y k A − |h x, e i A ||h y, e i A | ) ≥ . Therefore, |h x, y i A − h x, e i A h e, y i A | ≤ k x k A k y k A − |h x, e i A ||h y, e i A | = ⇒ |h x, e i A h e, y i A | − |h x, y i A | ≤ k x k A k y k A − |h x, e i A ||h e, y i A | . Hence, 2 |h x, e i A h e, y i A | ≤ |h x, y i A | + k x k A k y k A . ANIKET BHANJA, PINTU BHUNIA AND KALLOL PAUL
This completes the proof of the lemma. (cid:3)
Theorem 2.13.
Let T ∈ B A ( H ) . Then the following inequalities hold: ( i ) dw A ( T ) ≤ (cid:13)(cid:13)(cid:13) | T | A + ( | T | A ) ♯ A | T | A (cid:13)(cid:13)(cid:13) A , ( ii ) dw A ( T ) ≤ (cid:0) w A ( T ) + k T k A (cid:1) + k T k A . Proof.
Let x ∈ H with k x k A = 1. Then using Lemma 2.12 we get, |h T x, x i A | + k T x k A = |h T x, x i A h x, T x i A | + h| T | A x, x i A h x, | T | A x i A ≤
12 ( k T x k A + h T x, T x i A ) + 12 ( k| T | A x k A + h| T | A x, | T | A x i A )= h| T | A x, x i A + h ( | T | A ) ♯ A | T | A x, x i A = h ( | T | A + ( | T | A ) ♯ A | T | A ) x, x i A . Therefore, taking supremum over all A -unit vectors in H , we get the inequality (i).Again considering |h T x, x i A | = |h T x, x i A h x, T ♯ A x i A | and then using Lemma 2.12,we get the inequality (ii). (cid:3) Remark 2.14.
It is well-known that if T is A -normaloid then k T k A = k T k A .Therefore, it is easy to observe that both the inequalities in Theorem 2.13 becomesequality if T is A -normaloid.In the next theorem we obtain an upper bound for the A -Davis-Wielandt radiusof operators in B A ( H ). For this we need the following lemma which follows fromLemma 2.12. Lemma 2.15.
Let x, y, e ∈ H with k e k A = 1 . Then k x k A k y k A − |h x, y i A | ≥ |h x, e i A h e, y i A | ( k x k A k y k A − |h x, y i A | ) . Theorem 2.16.
Let T ∈ B A ( H ) . Then dw A ( T ) ≤ (cid:13)(cid:13)(cid:13) ( | T | A ) ♯ A | T | A + | T | A (cid:13)(cid:13)(cid:13) A − c A ( | T | A + T ) m A ( | T | A + T ) − c A ( | T | A − T ) m A ( | T | A − T ) . Proof.
Let x ∈ H with k x k A = 1. Then using Lemma 2.15 and Lemma 2.12 weget, |h T x, x i A | ≤ k T x k A k x k A − |h T x, x i A h x, x i A | ( k T x k A k x k A − |h T x, x i A | )= k T x k A + 2 |h T x, x i A ||h x, T x i A | − |h T x, x i A |k T x k A ≤ k T x k A + k T x k A + h T x, T x i A − c A ( T ) k T x k A ≤ h| T | A x, x i A − c A ( T ) m A ( T ) . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 9
Using the above inequality, we get |h T x, x i A | + k T x k A = 12 (cid:0) |k T x k A + h T x, x i A | + |k T x k A − h T x, x i A | (cid:1) = 12 (cid:0) |h ( | T | A + T ) x, x i A | + |h ( | T | A − T ) x, x i A | (cid:1) ≤ (cid:16) (cid:28)(cid:12)(cid:12)(cid:12) | T | A + T (cid:12)(cid:12)(cid:12) A x, x (cid:29) A − c A ( | T | A + T ) m A ( | T | A + T )+3 (cid:28)(cid:12)(cid:12)(cid:12) | T | A − T (cid:12)(cid:12)(cid:12) A x, x (cid:29) A − c A ( | T | A − T ) m A ( | T | A − T ) (cid:17) = 32 (cid:28)(cid:18)(cid:12)(cid:12)(cid:12) | T | A + T (cid:12)(cid:12)(cid:12) A + (cid:12)(cid:12)(cid:12) | T | A − T (cid:12)(cid:12)(cid:12) A (cid:19) x, x (cid:29) A − c A ( | T | A + T ) m A ( | T | A + T ) − c A ( | T | A − T ) m A ( | T | A − T )= 3 D(cid:16) ( | T | A ) ♯ A | T | A + | T | A (cid:17) x, x E A − c A ( | T | A + T ) m A ( | T | A + T ) − c A ( | T | A − T ) m A ( | T | A − T ) . Therefore, taking supremum over all A -unit vectors in H , we get the requiredinequality. (cid:3) Remark 2.17.
We would like to note that the inequality in [4, Th. 2.20] followsfrom Theorem 2.16 by considering A = I .Next we prove the following lemma. Lemma 2.18.
Let x, y ∈ H and λ ∈ C . Then we have the following equality: k x k A k y k A − |h x, y i A | = k x − λy k A k y k A − |h x − λy, y i A | . Proof.
We have, k x − λy k A k y k A − |h x − λy, y i A | = h x − λy, x − λy i A k y k A − |h x, y i A − λ k y k A | = (cid:0) k x k A + | λ | k y k A − Re ( λ h x, y i A ) (cid:1) k y k A − |h x, y i A | − | λ | k y k A +2 Re ( λ h x, y i A ) k y k A = k x k A k y k A − |h x, y i A | . (cid:3) Using Lemma 2.18, we obtain the following upper bound for the A -Davis-Wielandtradius of operator in B A ( H ) . Theorem 2.19.
Let T ∈ B A ( H ) . Then dw A ( T ) ≤ inf λ ∈ R sup θ ∈ R n | λ |k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) − | T | A + sin θIm A ( T ) k A o . In particular, dw A ( T ) ≤
12 sup θ ∈ R n (cid:13)(cid:13) cos θ Re A ( T ) + | T | A + sin θ Im A ( T ) (cid:13)(cid:13) A + (cid:13)(cid:13) cos θ Re A ( T ) − | T | A + sin θ Im A ( T ) (cid:13)(cid:13) A o . Proof.
Let x ∈ H with k x k A = 1 . Then there exists θ ∈ R such that |h T x, x i A | = e − i θ h T x, x i A . Using the Cartesian decomposition of T , i.e., T = Re A ( T )+i Im A ( T ) , we get, |h T x, x i A | = h e − i θ T x, x i A = h ((cos θ − i sin θ )( Re A ( T ) + i Im A ( T ))) x, x i A = h (cos θRe A ( T ) + sin θIm A ( T )) x, x i A + i h (cos θIm A ( T ) − sin θRe A ( T )) x, x i A . Since |h T x, x i A | ∈ R , |h T x, x i A | = h (cos θRe A ( T ) + sin θIm A ( T )) x, x i A . Now usingLemma 2.18, we get for any λ ∈ R , |h T x, x i A | = |h (cos θRe A ( T ) + sin θIm A ( T )) x, x i A | = k (cos θRe A ( T ) + sin θIm A ( T )) x k A −k (cos θRe A ( T ) + sin θIm A ( T )) x − λx k A + |h (cos θRe A ( T ) + sin θIm A ( T )) x − λx, x i A | A = h (cos θRe A ( T ) + sin θIm A ( T )) x, x i A −h (cos θRe A ( T ) + sin θIm A ( T ) − λI ) x, x i A + |h (cos θRe A ( T ) + sin θIm A ( T ) − λI ) x, x i A | = Dn (cos θRe A ( T ) + sin θIm A ( T )) − (cos θRe A ( T ) + sin θIm A ( T ) − λI ) o x, x E A + |h (cos θRe A ( T ) + sin θIm A ( T ) − λI ) x, x i A | = h (2 λ (cos θRe A ( T ) + sin θIm A ( T )) − λ I ) x, x i A + |h (cos θRe A ( T ) + sin θIm A ( T ) − λI ) x, x i A | . Similarly, using Lemma 2.18, we have k T x k A = |h| T | A x, x i A | = h (2 λ | T | A − λ I ) x, x i A + |h ( | T | A − λI ) x, x i A | . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 11
Now, |h T x, x i A | + k T x k A = h λ { cos θRe A ( T ) + | T | A + sin θIm A ( T ) } x, x i A − λ + 12 |h (cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI ) x, x i A | + 12 |h (cos θRe A ( T ) − | T | A + sin θIm A ( T )) x, x i A | ≤ | λ |k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) − | T | A + sin θIm A ( T ) k A ≤ sup θ ∈ R n | λ |k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) − | T | A + sin θIm A ( T ) k A o . Therefore, taking supremum over all A -unit vectors in H , we get dw A ( T ) ≤ sup θ ∈ R n | λ |k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) + | T | A + sin θIm A ( T ) − λI k A + 12 k cos θRe A ( T ) − | T | A + sin θIm A ( T ) k A o . This inequality holds for all λ ∈ R , so we get the desired inequality. In particular,if we choose λ = 0, then dw A ( T ) ≤
12 sup θ ∈ R n (cid:13)(cid:13) cos θ Re A ( T ) + | T | A + sin θ Im A ( T ) (cid:13)(cid:13) A + (cid:13)(cid:13) cos θ Re A ( T ) − | T | A + sin θ Im A ( T ) (cid:13)(cid:13) A o . (cid:3) Remark 2.20.
We would like to note that the inequality in [4, Th. 2.23] followsfrom Theorem 2.19 by considering A = I .Next we obtain the following inequality. Theorem 2.21.
Let T ∈ B A ( H ) . Then dw A ( T ) ≤ inf λ ∈ C n(cid:16) k Re ( λ ) Re A ( T ) + Im ( λ ) Im A ( T ) k A + (cid:13)(cid:13) | T | A − Re ( λT ) (cid:13)(cid:13) A (cid:17) +2 k Re ( λT ) k A − | λ | + w A ( T − λI ) o . In particular, dw A ( T ) ≤ p w A ( T ) + k T k A .Proof. Let x ∈ H with k x k A = 1 . Let λ ∈ C . Using Lemma 2.18 we get, k T x k A k x k A − |h T x, x i A | = k T x − λx k A k x k A − |h T x − λx, x i A | . Using Cartesian decomposition of T , i.e., T = Re A ( T ) + i Im A ( T ), we get, k T x k A = ( h Re A ( T ) x, x i A ) − ( h Re A ( T − λI ) x, x i A ) + ( h Im A ( T ) x, x i A ) − ( h Im A ( T − λI ) x, x i A ) + k T x − λx k A = h (2 Re A ( T ) − Re ( λ ) I ) x, x i A h Re ( λ ) x, x i A + h (2 Im A ( T ) − Im ( λ ) I ) x, x ih Im ( λ ) x, x i A + k T x − λx k A = 2 Re ( λ ) h Re A ( T ) x, x i A + 2 Im ( λ ) h Im A ( T ) x, x i A − ( Re ( λ )) − ( Im ( λ )) + k T x − λx k A = 2 ( Re ( λ ) h Re A ( T ) x, x i + Im ( λ ) h Im A ( T ) x, x i A ) − | λ | + h T x − λx, T x − λx i A = 2 ( Re ( λ ) h Re A ( T ) x, x i A + Im ( λ ) h Im A ( T ) x, x i A )+ (cid:10) ( | T | A − Re A ( λT )) x, x (cid:11) A ≤ k Re ( λ ) Re A ( T ) + Im ( λ ) Im A ( T ) k A + (cid:13)(cid:13) | T | A − Re A ( λT ) (cid:13)(cid:13) A . Again using Lemma 2.18 we get, |h T x, x i A | = k T x k A − k T x − λx k A + |h T x − λx, x i A | = 2 h Re ( λT ) x, x i A − | λ | + |h T x − λx, x i A | ≤ k Re A ( λT ) k − | λ | + w A ( T − λI ) . Hence, |h T x, x i A | + k T x k A ≤ k Re A ( λT ) k − | λ | + w A ( T − λI )+ (cid:0) k Re ( λ ) Re A ( T ) + Im ( λ ) Im A ( T ) k + (cid:13)(cid:13) | T | A − Re A ( λT ) (cid:13)(cid:13) A (cid:1) . Therefore, taking supremum over all A -unit vectors in H , and then taking infimumover all λ ∈ C , we get dw A ( T ) ≤ inf λ ∈ C n (cid:0) k Re ( λ ) Re A ( T ) + Im ( λ ) Im A ( T ) k A + (cid:13)(cid:13) | T | A − Re A ( λT ) (cid:13)(cid:13) A (cid:1) +2 k Re A ( λT ) k A − | λ | + w A ( T − λI ) o . Taking λ = 0, we get dw A ( T ) ≤ p w A ( T ) + k T k A . (cid:3) Remark 2.22.
We would like to note that the inequality in [4, Th. 2.24] followsfrom Theorem 2.21 by considering A = I .In the following theorem we obtain an upper bound for the A -Davis-Wielandt radiusof sum of two operators in B A ( H ) . Theorem 2.23.
Let
X, Y ∈ B A ( H ) . Then dw A ( X + Y ) ≤ dw A ( X ) + dw A ( Y ) + w A ( X ♯ A Y + Y ♯ A X ) . In particular, if A ( X ♯ A Y + Y ♯ A X ) = O then dw A ( X + Y ) ≤ dw A ( X ) + dw A ( Y ) . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 13
Proof.
From the definition of the A-Davis-Wielandt shell we get, DW A ( X + Y ) = n(cid:16) h ( X + Y ) x, x i A , h ( X + Y ) x, ( X + Y ) x i A (cid:17) : x ∈ H , k x k A = 1 o = n(cid:16) h Xx, x i A , h Xx, Xx i A (cid:17) + (cid:16) h Y x, x i A , h Y x, Y x i A (cid:17) + (cid:16) , h ( X ♯ A Y + Y ♯ A X ) x, x i A (cid:17) : x ∈ H , k x k A = 1 o . Hence, DW A ( X + Y ) ⊆ DW A ( X ) + DW A ( Y ) + A , where A = (cid:8)(cid:0) , h ( X ♯ A Y + Y ♯ A X ) x, x i A (cid:1) : x ∈ H , k x k A = 1 (cid:9) . This implies the first inequality of the theorem. In particular, if we consider A ( X ♯ A Y + Y ♯ A X ) = O, then we get the second inequality. (cid:3) Remark 2.24.
If we consider A = I in Theorem 2.23 then we get the inequalitiesin [4, Lemma 3.3 and Prop. 3.4].Next we state the following lemma, proof of which can be found in [6, Lemma 3.1]. Lemma 2.25.
Let T ij ∈ B A ( H ) , for i, j = 1 , . Then ( T ij ) × ∈ B A ( H ⊕ H ) and (cid:18) T T T T (cid:19) ♯ A = (cid:18) T ♯ A T ♯ A T ♯ A T ♯ A (cid:19) . Using Theorem 2.23 and Lemma 2.25, we prove the following inequality.
Corollary 2.26.
Let
X, Y ∈ B A ( H ), then dw A (cid:18) O XY O (cid:19) ≤ r k X k A + k X k A + r k Y k A + k Y k A . Proof.
Clearly, (cid:18)
O XO O (cid:19) ♯ A (cid:18) O OY O (cid:19) + (cid:18) O OY O (cid:19) ♯ A (cid:18) O XO O (cid:19) = (cid:18) O OO O (cid:19) .Therefore, from Theorem 2.23, we get, dw A (cid:18) O XY O (cid:19) ≤ dw A (cid:18) O XO O (cid:19) + dw A (cid:18) O OY O (cid:19) ≤ s w A (cid:18) O XO O (cid:19) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O XO O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A + s w A (cid:18) O OY O (cid:19) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O OY O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A = s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O XO O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O XO O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A + s (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O OY O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) O OY O (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) A , as A (cid:18) O XO O (cid:19) = A (cid:18) O OY O (cid:19) = (cid:18) O OO O (cid:19) , see [9, Cor. 2.2]= r k X k A + k X k A + r k Y k A + k Y k A , by using [5, Remark 3] . (cid:3) Remark 2.27.
In particular, if we consider A = I in Corollary 2.26 then we havethe inequality in [4, Th. 3.5]. Next we state the following lemma, proof of which follows from DW A ( U ♯ A T U ) = DW A ( T ), where T ∈ B A / ( H ) and U ∈ B A ( H ) is an A -unitary operator. Lemma 2.28.
Let T ∈ B A / ( H ) and U ∈ B A ( H ) be an A -unitary operator. Then dw A ( U ♯ A T U ) = dw A ( T ) . Using Lemma 2.28, we prove the following lemma.
Lemma 2.29.
Let
X, Y ∈ B A / ( H ) . Then(a) dw A (cid:18) O Xe i θ Y O (cid:19) = dw A (cid:18) O XY O (cid:19) , for every θ ∈ R . (b) dw A (cid:18) O XY O (cid:19) = dw A (cid:18) O YX O (cid:19) . Proof. (a) Let U = (cid:18) I OO e i θ I (cid:19) . Then using Lemma 2.28 we get, dw A (cid:18) O Xe i θ Y O (cid:19) = dw A (cid:18) U ♯ A (cid:18) O Xe i θ Y O (cid:19) U (cid:19) = dw A O e i θ Xe i θ Y O ! = dw A (cid:18) O XY O (cid:19) . (b) Considering U = (cid:18) O II O (cid:19) and using Lemma 2.28, we get (b). (cid:3)
Using Lemma 2.29, we obtain an upper bound for the A -Davis-Wielandt radius ofsum of product operators in B A ( H ) . Theorem 2.30.
Let
P, Q, X, Y ∈ B A ( H ) . Then for any t ∈ R \ { } , we have dw A ( P XQ ♯ A ± QY P ♯ A ) ≤ (cid:18) t k P k A + 1 t k Q k A (cid:19) ((cid:18) t k P X k A + 1 t k QY k A (cid:19) + α ) , where α = w A (cid:18) O XY O (cid:19) . Proof.
Let
C, Z ∈ B A ( H ⊕ H ) be such that C = (cid:18) P QO O (cid:19) and Z = (cid:18) O XY O (cid:19) .Then we have,
CZC ♯ A = (cid:18) P XQ ♯ A + QY P ♯ A OO O (cid:19) . Therefore, dw A ( P XQ ♯ A + QY P ♯ A ) ≤ dw A (cid:18) P XQ ♯ A + QY P ♯ A OO O (cid:19) = dw A ( CZC ♯ A )= sup k x k A =1 (cid:8) |h CZC ♯ A x, x i A | + k CZC ♯ A x k A (cid:9) = sup k x k A =1 (cid:8) |h ZC ♯ A x, C ♯ A x i A | + k CZC ♯ A x k A (cid:9) ≤ sup k x k A =1 (cid:8) w A ( Z ) k C ♯ A x k A + k CZ k A k C ♯ A x k A (cid:9) = (cid:0) w A ( Z ) + k CZ k A (cid:1) k C k A . N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 15
It is easy to see that k C k A = k P P ♯ A + QQ ♯ A k A and k CZ k A = k ( QY )( QY ) ♯ A +( P X )( P X ) ♯ A k A . Therefore, from the above inequality, we get dw A ( P XQ ♯ A + QY P ♯ A ) ≤ (cid:0) k P k A + k Q k A (cid:1) (cid:8) ( k QY k A + k P X k A ) + w A ( Z ) (cid:9) . Replacing Y by − Y in the above inequality and using Lemma 2.29 (a), we get dw A ( P XQ ♯ A − QY P ♯ A ) ≤ (cid:0) k P k A + k Q k A (cid:1) (cid:8) ( k QY k A + k P X k A ) + w A ( Z ) (cid:9) . Clearly, the above two inequalities hold for all
P, Q ∈ B A ( H ) . So, replacing P by tP and Q by t Q , we get the required inequality of the theorem. (cid:3) Corollary 2.31.
Let
P, Q, X, Y ∈ B A ( H ) with k P k A , k Q k A = 0 . then( i ) dw A ( P XQ ♯ A ± QY P ♯ A ) ≤ k P k A k Q k A n(cid:16) k P k A k Q k A k QY k A + k Q k A k P k A k P X k A (cid:17) + α o , where α = w A (cid:18) O XY O (cid:19) . ( ii ) dw A ( X ± Y ) ≤ (cid:26)(cid:0) k X k A + k Y k A (cid:1) + w A (cid:18) O XY O (cid:19)(cid:27) . Proof.
Considering t = q k Q k A k P k A in Theorem 2.30, we get the inequality (i). Choos-ing P = Q = I in (i), we get the inequality (ii). (cid:3) Corollary 2.32.
Let
P, Q, X, Y ∈ B A ( H ) be such that k P X k A , k QY k A = 0 . Then( i ) dw A ( P XQ ♯ A ± QY P ♯ A ) ≤ (cid:18) k QY k A k P X k A k P k A + k P X k A k QY k A k Q k A (cid:19) (cid:8) k P X k A k QY k A + α (cid:9) , where α = w A (cid:18) O XY O (cid:19) . ( ii ) dw A ( X ± Y ) ≤ (cid:18) k Y k A k X k A + k X k A k Y k A (cid:19) (cid:26) (2 k X k A k Y k A ) + w A (cid:18) O XY O (cid:19)(cid:27) . Proof.
Considering t = q k QY k A k P X k A in Theorem 2.30, we get the inequality (i). Choos-ing P = Q = I in (i), we get the inequality (ii). (cid:3) Remark 2.33.
Feki in [11, Prop. 3] proved that if
X, Y ∈ B A / ( H ) then thefollowing inequality holds: dw A ( X + Y ) ≤ (cid:16) dw A ( X ) + dw A ( Y ) (cid:17) + 4 (cid:16) dw A ( X ) + dw A ( Y ) (cid:17) . If we consider A = (cid:18) (cid:19) , X = (cid:18) (cid:19) and Y = (cid:18) (cid:19) then [11, Prop.3] gives dw A ( X + Y ) ≤ . dw A ( X + Y ) ≤ . dw A ( X + Y ) ≤ . dw A ( X + Y ) ≤ . A -Davis-Wielandt radius of special typeof 2 × B A / ( H ⊕ H ). Theorem 2.34.
Let X ∈ B A / ( H ) and T = (cid:18) I XO O (cid:19) . Then dw A ( T ) = ( √ , k X k A = 0(cos θ + k X k A sin θ )(cos θ + (cos θ + k X k A sin θ ) ) , k X k A = 0 , where b = k X k A , p = − b − b , q = − b − b , r = − b , s = b (8 b + 20 b +45 b + 61 b + 28) , α = (2 p − pq + 27 r ) , β = ( − α + √ s ) , γ = ( − α − √ s ) and θ = tan − ( β + γ − p ) . Proof.
Let z = (cid:18) xy (cid:19) ∈ H ⊕ H be such that k z k A = 1, i.e, k x k A + k y k A = 1. Then h T z, z i A = h x + Xy, x i A and h T z, T z i A = h x + Xy, x + Xy i A . Now, we have |h T z, z i A | + |h T z, T z i A | ≤ k x + Xy k A k x k A + k x + Xy k A = k x + Xy k A (cid:0) k x k A + k x + Xy k A (cid:1) ≤ sup k x k A + k y k A =1 ( k x k A + k X k A k y k A ) ( k x k A + ( k x k A + k X k A k y k A ) )= sup θ ∈ [0 , π ] (cos θ + k X k A sin θ ) (cos θ + (cos θ + k X k A sin θ ) ) . First we consider the case k X k A = 0 . Thensup θ ∈ [0 , π ] (cos θ + k X k A sin θ ) (cos θ + (cos θ + k X k A sin θ ) ) = 2 . Therefore, dw A ( T ) ≤ √ . Now let z = (cid:18) x (cid:19) be such that k z k A = 1, i.e., k x k A = 1.Then h T z, z i A = k x k A and h T z, T z i A = k x k A . Hence, ( |h T z, z i A | + |h T z, T z i A | ) = √ . Therefore, dw A ( T ) = √ . Next we consider the case k X k A = 0 . Thensup θ ∈ [0 , π ] (cos θ + k X k A sin θ ) (cos θ + (cos θ + k X k A sin θ ) )= (cos θ + k X k A sin θ ) (cos θ + (cos θ + k X k A sin θ ) ) , where b = k X k A , p = − b − b , q = − b − b , r = − b , s = b (8 b + 20 b +45 b + 61 b + 28) , α = (2 p − pq + 27 r ) , β = ( − α + √ s ) , γ = ( − α − √ s ) and θ = tan − ( β + γ − p ) . Therefore, dw A ( T ) ≤ (cos θ + k X k A sin θ )(cos θ + (cos θ + k X k A sin θ ) ) . We now show that there exists a sequence { z n } in H ⊕ H with k z n k A = 1 such thatlim n →∞ ( |h T z n , z n i A | + |h T z n , T z n i A | ) = (cos θ + k X k A sin θ )(cos θ + (cos θ + k X k A sin θ ) ) . Since X ∈ B A / ( H ), there exists a sequence { y n } in H with k y n k A = 1 such that lim n →∞ k Xy n k A = k X k A . Let z kn = √ k Xy n k A + k (cid:18) Xy n ky n (cid:19) ,where k ≥
0. Then |h T z kn , z kn i A | + |h T z kn , T z kn i A | = (1+ k ) k Xy n k A ( k Xy n k A + k ) (cid:0) k ) (cid:1) N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 17 = (cid:18) k Xy n k A √ k Xy n k A + k + k k Xy n k A √ k Xy n k A + k (cid:19) k Xy n k A k Xy n k A + k + (cid:18) k Xy n k A √ k Xy n k A + k + k k Xy n k A √ k Xy n k A + k (cid:19) ! . We can choose k ≥ k X k A √ k X k A + k = cos θ and k √ k X k A + k = sin θ .Therefore, if we choose z n = √ k Xy n k A + k (cid:18) Xy n k y n (cid:19) , then lim n →∞ ( |h T z n , z n i A | + |h T z n , T z n i A | ) = (cid:16) cos θ + k X k A sin θ (cid:17)(cid:16) cos θ +(cos θ + k X k A sin θ ) (cid:17) . Thiscompletes the proof. (cid:3)
Theorem 2.35.
Let X ∈ B A / ( H ) and S = (cid:18) O XO O (cid:19) . Then dw A ( S ) = , k X k A = 0 k X k A √ −k X k A , k X k A < √ k X k A , k X k A ≥ √ . Proof.
Let z = (cid:18) xy (cid:19) ∈ H ⊕ H be such that k z k A = 1, i.e, k x k A + k y k A = 1. Then h S z, z i A = h Xy, x i A and h S z, S z i A = h Xy, Xy i A . Now we have |h S z, z i A | + |h S z, S z i A | ≤ k Xy k A k x k A + k Xy k A ≤ sup k x k A + k y k A =1 (cid:0) k X k A k y k A k x k A + k X k A k y k A (cid:1) = sup θ ∈ [0 , π ] k X k A sin θ (cid:0) cos θ + k X k A sin θ (cid:1) . First we consider the case k X k A = 0 . Then it is easy to see that dw A ( S ) = 0 . Next we consider the case 0 < k X k A < √ . Thensup θ ∈ [0 , π ] k X k A sin θ (cid:0) cos θ + k X k A sin θ (cid:1) = k X k A − k X k A ) . Therefore, dw A ( S ) ≤ k X k A √ (1 −k X k A ) . We now show that there exists a sequence { z n } in H ⊕ H with k z n k A = 1 such thatlim n →∞ {|h S z n , z n i A | + |h S z n , S z n i A | } = k X k A p (1 − k X k A ) . Since X ∈ B A / ( H ), there exists a sequence { y n } in H with k y n k A = 1 such thatlim n →∞ k Xy n k A = k X k A . Let z n = √ k Xy n k A + k (cid:18) Xy n ky n (cid:19) , where k = k X k A √ − k X k A .Then lim n →∞ {|h S z n , z n i A | + |h S z n , S z n i A | } = k X k A p − k X k A . Therefore, dw A ( S ) = k X k A √ (1 −k X k A ) . Now we consider the case k X k A ≥ √ . Thensup θ ∈ [0 , π ] k X k A sin θ (cid:0) cos θ + k X k A sin θ (cid:1) = k X k A . Therefore, dw A ( S ) ≤ k X k A . We now show that there exists a sequence { z n } in H ⊕ H with k z n k A = 1 such thatlim n →∞ ( |h S z n , z n i A | + |h S z n , S z n i A | ) = k X k A . Since X ∈ B A / ( H ), there exists a sequence { y n } in H with k y n k A = 1 suchthat lim n →∞ k Xy n k A = k X k A . If we consider z n = (cid:18) y n (cid:19) , then h S z n , z n i A = 0and h S z n , S z n i A = k Xy n k A . Therefore, lim n →∞ ( |h S z n , z n i A | + |h S z n , S z n i A | ) = k X k A . This completes the proof. (cid:3)
Remark 2.36.
In particular, if we consider A = I in Theorem 2.34 and Theorem2.35 then we get [4, Th. 3.14] and [4, Th. 3.16], respectively. References
1. M.L. Arias, G. Corach and M.C. Gonzalez, Partial isometries in semi-Hilbertian spaces, LinearAlgebra Appl. 428 (7) (2008) 1460-1475.2. M.L. Arias, G. Corach and M.C. Gonzalez, Metric properties of projections in semi-Hilbertianspaces, Integral Equations Operator Theory 62 (2008) 11-28.3. H. Baklouti, K.Feki, O.A.M. Sid Ahmed, Joint normality of operators in semi-Hilbertianspaces, Linear Multilinear Algebra 68(4) (2020) 845-866.4. P. Bhunia, A. Bhanja, S. Bag and K. Paul, Bounds for the Davis-Wielandt radius of boundedlinear operators, http://arxiv.org/abs/2006.04389 .5. P. Bhunia, R.K. Nayak and K. Paul, Refinements of A-numerical radius inequalities and theirapplications, Adv. Oper. Theory (2020). https://doi.org/10.1007/s43036-020-00056-8 .6. P. Bhunia, K. Feki and K. Paul, A-Numerical radius orthogonality and parallelism ofsemi-Hilbertian space operators and their applications, Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00392-8 .7. C. Davis, The shell of a Hilbert-space operator, Acta Sci. Math., (Szeged) 29 (1968) 69-86.8. R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space,Proc. Amer. Math. Soc. 17 (1966) 413-416.9. K. Feki, Spectral radius of semi-Hilbertian space operators and its applications, Ann. Funct.Anal. (2020). https://doi.org/10.1007/s43034-020-00064-y .10. K. Feki, Inequalities for the A-joint numerical radius of two operators and their applications,arXiv:2005.04758v1 [math.FA] 10 May (2020).11. K. Feki and O.A.M. Sid Ahmed, Davis-Wielandt shells of semi-Hilbertianspace operators and its applications, Banach J. Math. Anal. (2020), https://doi.org/10.1007/s43037-020-00063-0 .12. C.-K. Li and Y.-T. Poon, Spectrum, numerical range and Davis-Wielandt Shell of a normaloperator, Glasgow Math. J. 51 (2009) 91-100.13. C.-K. Li, Y.-T. Poon and N.S. Sze, Davis-Wielandt shells of operators, Oper. Matrices, 2(3)(2008) 341-355, https://dx.doi.org/10.7153/oam-02-20 .14. B. Lins, I.M. Spitkovsky, S. Zhong, The normalized numerical range and the Davis-Wielandtshell, Linear Algebra Appl. 546 (2018) 187-209.15. A. Zamani, A-numerical radius inequalities for semi-Hilbertian space operators, Linear Alge-bra Appl. 578 (2019) 159-183.16. A. Zamani and K. Shebrawi, Some Upper Bounds for the Davis-Wielandt Radius of HilbertSpace Operators, Mediterr. J. Math., (2019), https://doi.org/10.1007/s00009-019-1458-z .17. A. Zamani, M.S. Moslehian, M.-T. Chien and H. Nakazato, Norm-parallelism and the Davis-Wielandt radius of Hilbert space operators, Linear Multilinear Algebra, 67(11), (2019) 2147-2158.18. H. Wielandt, On eigenvalues of sums of normal matrices, Pac. J. Math. 5 (1955) 633-638.(Bhanja)
Department of Mathematics, Vivekananda College Thakurpukur, Kolkata,West Bengal, India
E-mail address : [email protected] N GENERALIZED DAVIS-WIELANDT RADIUS INEQUALITIES 19 (Bhunia)
Department of Mathematics, Jadavpur University, Kolkata 700032, WestBengal, India
E-mail address : [email protected] (Paul) Department of Mathematics, Jadavpur University, Kolkata 700032, West Ben-gal, India
E-mail address ::