A -Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications
aa r X i v : . [ m a t h . F A ] J a n A -NUMERICAL RADIUS ORTHOGONALITY ANDPARALLELISM OF SEMI-HILBERTIAN SPACE OPERATORSAND THEIR APPLICATIONS PINTU BHUNIA , KAIS FEKI AND KALLOL PAUL Abstract.
In this paper, we aim to introduce and characterize the conceptof numerical radius orthogonality of operators on a complex Hilbert space H which are bounded with respect to the semi-norm induced by a positiveoperator A on H . Moreover, a characterization of the A -numerical radiusparallelism for A -rank one operators is proved. As applications of the obtainedresults, we obtain some A -numerical radius inequalities of operator matriceswhere A is the operator diagonal matrix with diagonal entries are positiveoperator A . Some other related results are also investigated. Introduction and Preliminaries
Throughout this paper, B ( H ) denote the C ∗ -algebra of all bounded linear oper-ators acting on a complex Hilbert space H with an inner product h· | ·i and thecorresponding norm k · k . The symbol I stands for the identity operator on H . Inall that follows, by an operator we mean a bounded linear operator. The rangeof every operator T is denoted by R ( T ), its null space by N ( T ) and T ∗ is theadjoint of T . Let B ( H ) + be the cone of positive (semi-definite) operators, i.e. B ( H ) + = { A ∈ B ( H ) : h Ax | x i ≥ , ∀ x ∈ H } . Every A ∈ B ( H ) + defines thefollowing positive semi-definite sesquilinear form: h· | ·i A : H × H −→ C , ( x, y ) x | y i A = h Ax | y i . Clearly, the induced semi-norm is given by k x k A = h x | x i / A , for every x ∈ H .This makes H into a semi-Hilbertian space. One can verify that k · k A is a normon H if and only if A is injective, and that ( H , k · k A ) is complete if and only if R ( A ) is closed. Definition 1.1. ([2]) Let A ∈ B ( H ) + and T ∈ B ( H ). An operator S ∈ B ( H ) iscalled an A -adjoint of T if for every x, y ∈ H , the identity h T x | y i A = h x | Sy i A holds. That is S is solution in B ( H ) of the equation AX = T ∗ A . Date : January 15, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Positive operator, numerical radius, orthogonality, parallelism, A -rank one operator.Pintu Bhunia would like to thank UGC, Govt. of India for the financial support in the formof SRF. Prof. Kallol Paul would like to thank RUSA 2.0, Jadavpur University for the partialsupport. The existence of an A -adjoint operator is not guaranteed. The set of all operatorswhich admit A -adjoints is denoted by B A ( H ). By Douglas Theorem [13], we have B A ( H ) = { T ∈ B ( H ) : R ( T ∗ A ) ⊆ R ( A ) } = { T ∈ B ( H ) : ∃ λ > such that k AT x k ≤ λ k Ax k , ∀ x ∈ H} . (1.1)If T ∈ B A ( H ), the reduced solution of the equation AX = T ∗ A is a distinguished A -adjoint operator of T , which is denoted by T ♯ A . Note that, T ♯ A = A † T ∗ A inwhich A † is the Moore-Penrose inverse of A . For more results concerning T ♯ A ,see [2, 3]. Again, by applying Douglas theorem, it can observed that B A / ( H ) = { T ∈ B ( H ) : ∃ λ > such that k T x k A ≤ λ k x k A , ∀ x ∈ H} . (1.2)Operators in B A / ( H ) are called A -bounded. Further, h· | ·i A induces the follow-ing semi-norm on B A / ( H ): k T k A = sup x ∈R ( A ) ,x =0 k T x k A k x k A = sup {k T x k A ; x ∈ H , k x k A = 1 } < + ∞ . (1.3)For the rest of this paper, A denotes a nonzero operator in B ( H ) + and P A willbe denoted to be the projection onto R ( A ). Moreover, it is important to pointout the following facts. The semi-inner product h· | ·i A induces an inner producton the quotient space H / N ( A ) defined as[ x, y ] = h Ax | y i , for all x, y ∈ H / N ( A ). Notice that ( H / N ( A ) , [ · , · ]) is not complete unless R ( A )is not closed. However, a canonical construction due to L. de Branges and J.Rovnyak in [12] shows that the completion of H / N ( A ) under the inner prod-uct [ · , · ] is isometrically isomorphic to the Hilbert space R ( A / ) with the innerproduct ( A / x, A / y ) = h P A x | P A y i , ∀ x, y ∈ H . In the sequel, the Hilbert space (cid:0) R ( A / ) , ( · , · ) (cid:1) will be denoted by R ( A / ) andwe use the symbol k · k R ( A / ) to represent the norm induced by the inner product( · , · ). The interested reader is referred to [4] for more information related to theHilbert space R ( A / ). Notice that the fact R ( A ) ⊂ R ( A / ) implies that( Ax, Ay ) = h x | y i A . (1.4)This leads to the following useful relation: k Ax k R ( A / ) = k x k A , ∀ x ∈ H . (1.5)The following useful proposition is taken from [4]. Proposition 1.2.
Let T ∈ B ( H ) . Then T ∈ B A / ( H ) if and only if there existsa unique e T ∈ B ( R ( A / )) such that Z A T = e T Z A . Here, Z A : H → R ( A / ) isdefined by Z A x = Ax . Before we move on, it is important to state the following useful lemmas.
Lemma 1.3. ([14]) If T ∈ B A / ( H ) , then we have -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 3 (1) k T k A = k e T k B ( R ( A / )) . (2) ω A ( T ) = ω ( e T ) ,where ω ( e T ) is the numerical radius of e T and ω A ( T ) is the A -numerical radius of T , defined as below. Lemma 1.4. ([17, Proposition 2.9])
Let T ∈ B A ( H ) . Then, g T ♯ A = (cid:0) e T (cid:1) ∗ and ^ ( T ♯ A ) ♯ A = e T .
Now we note 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. Foran account of results, we refer to [2, 3, 14].It is useful to recall that an operator T is called A -self-adjoint if AT is self-adjoint (i.e. AT = T ∗ A ) and it is called A -positive if AT ≥ H were extended when an additional semi-inner product defined by A ∈ B ( H ) + isconsidered. One may see [7, 8, 10, 11, 17, 22].The generalization of the numerical range, known as A -numerical range (see [9])is given by: W A ( T ) = {h T x | x i A : x ∈ H , k x k A = 1 } . The A -numerical radius ω A ( T ) and the A -Crawford number m A ( T ) of an operator T are defined as: ω A ( T ) = sup {| λ | : λ ∈ W A ( T ) } ,m A ( T ) = inf {| λ | : λ ∈ W A ( T ) } . It is well-known that the A -numerical radius of an A -bounded operator T isequivalent to A -operator semi-norm of T , (see [22]). More precisely, we have k T k A ≤ ω A ( T ) ≤ k T k A . (1.6)Zamani [22] studied A -numerical radius inequalities for semi-Hilbertian spaceoperators. In [7, 8], we have also studied A -numerical radius inequalities of d × d operator matrices where A is the d × d diagonal operator matrix whose diagonalentries are A . Notice that the study of numerical radius inequalities receivedconsiderable attention in the last decades (the reader is invited to consult forexample [5, 6] and the references therein). An operator U ∈ B A ( H ) is said tobe A -unitary if U ♯ A U = ( U ♯ A ) ♯ A U ♯ A = P A . We mention that if T ∈ B A ( H ) then T ♯ A ∈ B A ( H ), ( T ♯ A ) ♯ A = P A T P A . Let T denote the unit cycle of the complexplane, i.e. T = { λ ∈ C : | λ | = 1 } . Recently, new types of parallelism for A -bounded operators based on the A -numerical radius and the A -operator semi-norm was introduced in [15]. More precisely, we have the following definition. Definition 1.5. ([15]) Let
T, S ∈ B A / ( H ).(1) We say that T is A -norm-parallel to S , in short T k A S , if there exists λ ∈ T such that k T + λS k A = k T k A + k S k A . PINTU BHUNIA, KAIS FEKI AND KALLOL PAUL (2) The operator T is said to be A -numerical radius parallel to S and wedenote T k ω A S , if ω A ( T + λS ) = ω A ( T ) + ω A ( S ) for some λ ∈ T . Also, the following theorems are proved in [15].
Theorem 1.6.
Let
T, S ∈ B A / ( H ) . Then, the following assertions are equiva-lent: (1) T k A S . (2) There exists a sequence ( x n ) n ⊂ H such that k x n k A = 1 , lim n → + ∞ |h T x n | Sx n i A | = k T k A k S k A . Theorem 1.7.
Let
T, S ∈ B A / ( H ) . Then the following conditions are equiva-lent: (1) T k ω A S . (2) There exists a sequence of A -unit vectors { x n } in H such that lim n → + ∞ (cid:12)(cid:12) h T x n | x n i A h Sx n | x n i A (cid:12)(cid:12) = ω A ( T ) ω A ( S ) . (1.7)Recently, Zamani introduced in [21] the notion of A -Birkhoff-James orthogonalityof operators in semi-Hilbertian spaces as follows. Definition 1.8. ([21]) An element T ∈ B A / ( H ) is called an A -Birkhoff-Jamesorthogonal to another element S ∈ B A / ( H ), denoted by T ⊥ BA S , if k T + γS k A ≥ k T k A for all γ ∈ C . The paper is organized as follows: In the next section, we introduce and give acharacterization of A -numerical radius orthogonality for A -bounded operators. Inparticular, for T, S ∈ B A / ( H ), we show that T is A -numerical radius orthogonalto S if and only if for each β ∈ [0 , π ), there exists a sequence of A -unit vectors { x k } in H such thatlim k → + ∞ |h T x k | x k i A | = ω A ( T ) and lim k → + ∞ ℜ e (cid:0) e iβ h x k | T x k i A h Sx k | x k i A (cid:1) ≥ . Furthermore, inspiring by the rank one operators in Hilbert spaces, we introducethe class of A -rank one operators in semi-Hilbert spaces. In addition, a charac-terization of the A -numerical radius parallelism of A -rank one operators is estab-lished. Our results cover and extend the works in [16, 18]. In the last section, wegive some inequalities for A -numerical radius of semi-Hilbertian space operatorswhich are as an application of A -numerical radius orthogonality and parallelism.The obtained results generalize and improve on the existing inequalities.2. A -numerical radius orthogonality and parallelism In this section, we introduce and completely characterize the concept of orthogo-nality of A -bounded operators with respect to the A -numerical radius ω ( · ). Alsowe give a characterization of A -numerical radius parallelism for A -rank one op-erators. First, let us introduce the notion of A -numerical radius orthogonality ofoperators in semi-Hilbertian spaces. -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 5 Definition 2.1.
An element T ∈ B A / ( H ) is called an A -numerical radius or-thogonal to another element S ∈ B A / ( H ), denoted by T ⊥ ω A S , if ω A ( T + γS ) ≥ ω A ( T ) for all γ ∈ C . In the following proposition we state some basic properties of A -numerical radiusorthogonality. The proof follows immediately from the definition of A -numericalradius orthogonality of operators and hence it is omitted. Proposition 2.2.
Let
T, S ∈ B A ( H ) . Then the following properties are equiva-lent. (i) T ⊥ ω A S . (ii) T ♯ A ⊥ ω A S ♯ A . (iii) αT ⊥ ω A βS for all α, β ∈ C \ { } . In the next proposition, we give some connections between A -numerical radiusorthogonality and A -Birkhoff-James orthogonality of operators. Recall from [14]that an operator T ∈ B A / ( H ) is said to be A -normaloid if r A ( T ) = k T k A , where r A ( T ) = lim n → + ∞ k T n k n A . Moreover, it was shown in [14] that an operator T ∈ B A / ( H ) is A -normaloid ifand only if ω A ( T ) = k T k A .Now we give the following proposition. Proposition 2.3.
Let
T, S ∈ B A / ( H ) . Then the following conditions hold: (1) If T is an A -normaloid operator, then T ⊥ ω A S ⇒ T ⊥ BA S . (2) If AT = 0 , then T ⊥ BA S ⇒ T ⊥ ω A S .Proof. (1) Notice first that since T is an A -normaloid operator, then ω A ( T ) = k T k A . Now, assume that T ⊥ ω A S . This implies that ω A ( T + λS ) ≥ ω A ( T ) forall λ ∈ C . Hence, by taking into account (1.6), we get k T + λS k A ≥ ω A ( T + λS ) ≥ ω A ( T ) = k T k A , for all λ ∈ C . Therefore, we deduce that T ⊥ BA S .(2) It was shown in [14, Corollary 2.2.] that if AT = 0 , then ω A ( T ) = k T k A .Now, we assume that T ⊥ BA S . Then, k T + λS k A ≥ k T k A for all λ ∈ C . Moreover,by using (1.6), we see that ω A ( T + λS ) ≥ k T + λS k A ≥ k T k A = ω A ( T ) , for all λ ∈ C . Thus, T ⊥ ω A S as required. (cid:3) Remark . In general, as it was point out in [16] that the above two notions oforthogonality are not equivalent.In the following theorem, we prove our first main result in this section, whichcharacterizes A -numerical radius orthogonality of A -bounded operators on com-plex Hilbert space. Theorem 2.5.
Let
T, S ∈ B A / ( H ) . Then, the following assertions are equiva-lent: PINTU BHUNIA, KAIS FEKI AND KALLOL PAUL (1) T ⊥ ω A S . (2) For each β ∈ [0 , π ) , there exists a sequence of A -unit vectors { x k } in H such that lim k → + ∞ |h T x k | x k i A | = ω A ( T ) and lim k → + ∞ ℜ e (cid:0) e iβ h x k | T x k i A h Sx k | x k i A (cid:1) ≥ . (2.1) Proof. (1) = ⇒ (2) : Assume that T ⊥ ω A S . Then, ω A ( T + λS ) ≥ ω A ( T ) , for every λ ∈ C . Let β ∈ [0 , π ). It follows from the definition of ω A ( · ) that, for every n ∈ N ∗ , there exists a sequence of A -unit vectors { z n } in H such that |h T z n + e iβ n Sz n | z n i A | > ω A ( T ) − n . (2.2)Therefore, for all n ∈ N ∗ we have( ω A ( T ) − n ) < |h T z n + e iβ n Sz n | z n i| . This implies that ω A ( T ) − n ω A ( T ) + n < |h T z n | z n i A | + n |h Sz n | z n i A | + n ℜ e (cid:0) e iβ h z n | T z n i A h Sz n | z n i A (cid:1) , which in turn yields that n (cid:2) ω A ( T ) − |h T z n | z n i A | (cid:3) < n ω A ( T ) − n + n |h Sz n | z n i A | + ℜ e (cid:0) e iβ h z n | T z n i A h Sz n | z n i A (cid:1) . Hence, we infer that n ω A ( T ) − n + n k S k A + ℜ e (cid:0) e iβ h z n | T z n i A h Sz n | z n i A (cid:1) > , (2.3)for all n ∈ N ∗ . Moreover, since T, S ∈ B A / ( H ) and k z n k A = 1, then by theCauchy-Schwarz inequality it can be seen that ( h T z n | z n i A ) n and ( h Sz n | z n i A ) n are bounded sequences of complex numbers. So, there exists a subsequence ( z n k ) k of ( z n ) n such thatlim k → + ∞ ℜ e (cid:0) e iβ h z n k | T z n k i A h Sz n k | z n k i A (cid:1) exists . Now, consider the sequence { x k } such that x k = z n k for all k . Clearly, k x k k A = 1for all k . Moreover, by using (2.3) we getlim k → + ∞ ℜ e (cid:0) e iβ h x k | T x k i A h Sx k | x k i A (cid:1) ≥ , as desired. On the other hand, by using the Cauchy-Schwarz inequality andtaking into consideration (2.2) together with the fact that n k ≥ k for all k , weobtain |h T x k | x k i A | ≥ |h T x k + e iβ k Sx k | x k i A | − k |h Sx k | x k i A | > ω A ( T ) − k − k k S k A . So, by letting k → + ∞ , we obtain lim k → + ∞ |h T x k | x k i A | ≥ ω A ( T ). This immedi-ately gives ω A ( T ) = lim k → + ∞ |h T x k | x k i A | , as required. Thus, the assertion (2) is proved. -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 7 (2) = ⇒ (1) : Let λ ∈ C . Then, there exists some β ∈ [0 , π ) such that λ = | λ | e iβ . So, by hypothesis, there exists a sequence of A -unit vectors { x k } in H such that (2.1) holds. Hence, we see that ω A ( T + λS ) ≥ lim k → + ∞ |h T x k + λSx k | x k i A | = lim k → + ∞ [ |h T x k | x k i A | + 2 | λ | ℜ e (cid:0) e iβ h x k | T x k i A h Sx k | x k i A (cid:1) + | λ | |h Sx k | x k i A | ] ≥ lim k → + ∞ |h T x k | x k i A | = ω A ( T ) . Thus, ω A ( T + λA ) ≥ ω A ( T ) for every λ ∈ C . Therefore, T ⊥ ω A S . Hence, theproof of the theorem is complete. (cid:3) We would like to emphasize the following remark.
Remark . For every β ∈ [0 , π ), the limit of ℜ e (cid:0) e iβ h z n | T z n i A h Sz n | z n i A (cid:1) need not exist in general for some sequence { z n } of A -unit vectors even if A = I .Indeed, let H = ℓ N ∗ ( C ) and ( e n ) n ∈ N ∗ be the canonical basis of ℓ N ∗ ( C ). Assumethat θ = 0 and A = T = I . Consider the following operator S : ℓ N ∗ ( C ) → ℓ N ∗ ( C ) , x = ( x n ) n ∈ N ∗ Sx = ( y n ) n ∈ N ∗ , such that y n = ( − n x n for all n ∈ N ∗ . Clearly, S ∈ B ( ℓ N ∗ ( C )). Moreover, ℜ e ( h Se n | e n i ) = ( − n . Now we aim to give a characterization of the A -numerical radius parallelismfor special type of semi-Hilbert space operators which will be called A -rank oneoperators. This new class of operators is defined as follows. Definition 2.7.
Let x, y ∈ H , the A -rank one operator is denoted by x ⊗ A y ,where x ⊗ A y is the following map: x ⊗ A y : H → H z ( x ⊗ A y )( z ) = h z | y i A x. In order to characterize of the A -numerical radius parallelism for A -rank oneoperators, we need the following two lemmas. Before that, it is useful to recallfor every x, y ∈ H , the rank one operator x ⊗ y verifies k x ⊗ y k = k x kk y k and ω ( x ⊗ y ) = ( |h x | y i| + k x kk y k ) . (2.4) Lemma 2.8.
Let x, y ∈ H . Then, the following properties hold. (1) k x ⊗ A y k A = k x k A k y k A . (2) ω A ( x ⊗ A y ) = ( h x | y i A + k x k A k y k A ) . PINTU BHUNIA, KAIS FEKI AND KALLOL PAUL
Proof. (1) Let z ∈ H . Then, by using the Cauchy-Schwarz inequality, we have k ( x ⊗ A y )( z ) k A = k A / ( x ⊗ A y )( z ) k = kh z | y i A A / x k = |h A / z | A / y i| × k x k A ≤ k x k A k y k A k z k A . Hence, x ⊗ A y ∈ B A / ( H ). So, by Proposition 1.2 there exists a unique ^ x ⊗ A y ∈B ( R ( A / )) such that Z A ( x ⊗ A y ) = ^ x ⊗ A yZ A . On the other hand, by using (1.4)we see that Z A ( x ⊗ A y )( z ) = h z | y i A Ax = ( Az, Ay ) Ax = ( Ax ⊗ Ay )( Az ) . for all z ∈ H . Hence, ^ x ⊗ A y = Ax ⊗ Ay , where Ax ⊗ Ay is defined as Ax ⊗ Ay : R ( A / ) → R ( A / ) z ( Ax ⊗ Ay )( z ) = ( z, Ay ) Ax.
So, by using (2.4) we obtain k x ⊗ A y k A = k ^ x ⊗ A y k B ( R ( A / )) = k Ax ⊗ Ay k B ( R ( A / )) = k Ax k R ( A / ) k Ay k R ( A / ) = k x k A k y k A . (2) By applying Lemma 1.3 together with (2.4) we get ω A ( x ⊗ A y ) = ω ( ^ x ⊗ A y )= ω ( Ax ⊗ Ay )= (cid:2) ( Ax, Ay ) + k Ax k R ( A / ) k Ay k R ( A / ) (cid:3) = ( h x | y i A + k x k A k y k A ) ( by (1.4) and (1.5)) . (cid:3) Remark . Very recently, as our work was in progress, the above lemma hasbeen proved by Zamani in [19]. Our proof here is different from his approach.
Lemma 2.10.
Let x, y ∈ H . Then, the following properties are equivalent. (1) x k A y ( i.e. k x + λy k A = k x k A + k y k A for some λ ∈ T ) . (2) x ⊗ A y k A I .Proof. By using (1.5), one can observe that x k A y if and only if Ax k Ay (thatis Ax and Ay are parallel on R ( A / )). On the other hand, it is well-known thatfor a given T, S ∈ B A / ( H ) we have T k A S if and only if e T k e S (see [15, Lemma3.1]). So, x ⊗ A y k A I if and only if ^ x ⊗ A y k e I . Since, ^ x ⊗ A y = Ax ⊗ Ay -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 9 and e I = I R ( A / ) , then we get the desired equivalence by applying [20, Corollary2.23]. (cid:3) Now, we have in a position to prove our second main result in this section.
Theorem 2.11.
Let x, y ∈ H . Then the following conditions are equivalent: (1) x k A y . (2) x ⊗ A x k ω A y ⊗ A y .Proof. (1) ⇒ (2) : Suppose that x k A y . Then, in view of Lemma 2.10, we have x ⊗ A y k A I . So, by Theorem 1.6 there exists a sequence of A -unit vectors { x n } in H such that lim n → + ∞ (cid:12)(cid:12) h ( x ⊗ A y ) x n | x n i A (cid:12)(cid:12) = k x ⊗ A y k A . (2.5)On the other hand, by Lemma 2.8 we have ω A ( x ⊗ A x ) ω A ( y ⊗ A y ) = k x k A k y k A = k x ⊗ A y k A . (2.6)Moreover, a short calculation shows that h ( x ⊗ A x ) x n | x n i A h ( y ⊗ A y ) x n | x n i A = (cid:12)(cid:12) h x | x n i A h x n | y i A (cid:12)(cid:12) = (cid:12)(cid:12) h ( x ⊗ A y ) x n | x n i A (cid:12)(cid:12) (2.7)So, by combining (2.5) together with (2.6) and (2.7), we getlim n → + ∞ (cid:12)(cid:12)(cid:12) h ( x ⊗ A x ) x n | x n i A h ( y ⊗ A y ) x n | x n i A (cid:12)(cid:12)(cid:12) = ω A ( x ⊗ A x ) ω A ( y ⊗ A y ) . (2.8)Therefore, by Theorem 1.7 we deduce that x ⊗ A x k ω A y ⊗ A y as required.(2) ⇒ (1) : Assume that x ⊗ A x k ω A y ⊗ A y . It follows from Theorem 1.7 thatthere exists a sequence of A -unit vectors { x n } in H such that (2.8) holds. So,by making the same computations as above and applying Theorem 1.6 togetherwith Lemma 2.10 we can easily show that x k A y . This completes the proof ofthe theorem. (cid:3) In the following theorem, we study the connection between A -numerical radiusparallelism and A -semi-norm parallelism. Theorem 2.12.
Let T , T ∈ B A ( H ) be A -normaloid operators. If T k ω A T ,then T k A T .Proof. Since T k ω A T , then there exists some λ ∈ T such that ω A ( T + λT ) = ω A ( T ) + ω A ( T ). In addition, since T and T are A -normaloid, then we have ω A ( T ) = k T k A and ω A ( T ) = k T k A . Therefore, we obtain k T k A + k T k A = ω A ( T ) + ω A ( T )= ω A ( T + λT ) ≤ k T + λT k A ≤ k T k A + k T k A . This implies that k T + λT k A = k T k A + k T k A for some λ ∈ T and so T k A T as required. (cid:3) Application: some A -numerical radius inequalities In this section, we present some applications of the A -numerical radius parallelismand the A -numerical radius orthogonality. In particular, we will prove someinequalities for the A -numerical radius of semi-Hilbertian space operators. Inorder to achieve the goals of this section, we need some results.In all what follows, we consider the Hilbert space H = ⊕ di =1 H equipped withthe following inner-product: h x, y i = d X k =1 h x k | y k i , for all x = ( x , · · · , x d ) ∈ H and y = ( y , · · · , y d ) ∈ H . Let A be a d × d operatordiagonal matrix with diagonal entries are the positive operator A , i.e. A = A · · · A · · · · · · A ∈ B ( H ) + . Then, A defines the following positive semi-definite sesquilinear form h x, y i A = h A x, y i = d X k =1 h Ax k | y k i = d X k =1 h x k | y k i A , for all x = ( x , · · · , x d ) , y = ( y , · · · , y d ) ∈ H . We begin with the followinglemma. Lemma 3.1.
Let T = ( T ij ) d × d be such that T ij ∈ B A ( H ) for all i, j . Then, T ∈ B A ( H ) . Moreover, we have T T · · · T d T T · · · T d ... ... ... ... T d T d · · · T dd ♯ A = T ♯ A T ♯ A · · · T ♯ A d T ♯ A T ♯ A · · · T ♯ A d ... ... ... ... T ♯ A d T ♯ A d · · · T ♯ A dd . (3.1) Proof.
By taking into account (1.1), we need to show that there exists λ > k AT x k ≤ λ k A x k , for all x = ( x , · · · , x d ) ∈ H or equivalently d X k =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X j =1 AT kj x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ λ d X k =1 k Ax k k . Let x = ( x , · · · , x d ) ∈ H . Since T ij ∈ B A ( H ) for all i, j , then by (1.1) thereexists µ ij > k AT ij x k ≤ µ ij k Ax k , -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 11 for all x ∈ H and i, j ∈ { , · · · , d } . So, we get k AT x k = d X k =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X j =1 AT kj x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ d X k =1 d X j =1 k AT kj x j k ! ≤ d X k =1 d X j =1 µ kj k Ax j k ! ≤ d (max k,j { µ kj } ) d X j =1 k Ax j k ! ≤ d (max k,j { µ kj } ) d X j =1 k Ax j k = λ k A x k , where λ = d ( max k,j { µ kj } ). Hence, T ∈ B A ( H ). In order to prove (3.1), let S = T ♯ A T ♯ A · · · T ♯ A d T ♯ A T ♯ A · · · T ♯ A d ... ... ... ... T ♯ A d T ♯ A d · · · T ♯ A dd . By using the fact that T ij ∈ B A ( H ) for all i, j ∈ { , · · · , d } , one obtains AS = AT ♯ A AT ♯ A · · · AT ♯ A d AT ♯ A AT ♯ A · · · AT ♯ A d ... ... ... ... AT ♯ A d AT ♯ A d · · · AT ♯ A dd = T ∗ A T ∗ A · · · T ∗ d AT ∗ A T ∗ A · · · T ∗ d A ... ... ... ... T ∗ d A T ∗ d A · · · T ∗ dd A = T T · · · T d T T · · · T d ... ... ... ... T d T d · · · T dd ∗ A . Finally, in order to get (3.1), we shall need to show that R ( S ) ⊆ R ( A ). Let x = ( x , x , · · · , x d ) ∈ H be arbitrary. Then S x = P dj =1 T ♯ A j x j P dj =1 T ♯ A j x j ... P dj =1 T ♯ A jd x j . Since T ♯ A jk x j ∈ R ( A ) , so P dj =1 T ♯ A jk x j ∈ R ( A ) for each k = 1 , , . . . , d . Thisimplies that S x ∈ R ( A ) ⊕ R ( A ) . . . ⊕ R ( A ) = R ( A ) . This completes the proof ofthe theorem. (cid:3)
Lemma 3.2.
Let T = ( T ij ) d × d be such that T ij ∈ B A / ( H ) for all i, j . Then, T ∈ B A / ( H ) . Moreover, we have e T = ( f T ij ) d × d . Proof.
By proceeding as in the proof of Lemma 3.1, we can show that there exists λ > k T x k A ≤ λ k x k A , for all x = ( x , · · · , x d ) ∈ H . Hence, by (1.2) we deduce that T ∈ B A / ( H ). So,by Proposition 1.2 there exists a unique e T ∈ B ( R ( A / )) such that Z A T = e T Z A .On the other hand, for all x = ( x , · · · , x d ) ∈ H we have Z A T x = A · · · A · · · · · · A P dj =1 T j x j P dj =1 T j x j ... P dj =1 T dj x j = P dj =1 AT j x j P dj =1 AT j x j ... P dj =1 AT dj x j . Since, T ij ∈ B A / ( H ) for all i ∈ { , · · · , d } . Then, by Proposition 1.2 there exists f T ij ∈ B ( R ( A / )) such that AT ij = f T ij A for all i ∈ { , · · · , d } . So, we get Z A T x = P dj =1 f T j Ax j P dj =1 f T j Ax j ... P dj =1 f T dj Ax j = f T f T · · · f T d f T f T · · · f T d ... ... ... ... f T d f T d · · · f T dd A · · · A · · · · · · A x x ... x d = f T f T · · · f T d f T f T · · · f T d ... ... ... ... f T d f T d · · · f T dd Z A x. -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 13 Hence, we infer that e T = ( f T ij ) d × d . (cid:3) The following refinement of the triangle inequality has been proved by F. Kit-taneh et al. in [1].
Theorem 3.3.
Let
T, S ∈ B ( H ) . Then, k T + S k ≤ ω (cid:18) TS ∗ (cid:19) ≤ k T k + k S k . Now, we extend the above theorem as follows.
Theorem 3.4.
Let T , T ∈ B A ( H ) and A = (cid:18) A A (cid:19) ∈ B ( H ⊕ H ) + . Then k T + T k A ≤ ω A (cid:18) O T T ♯ A O (cid:19) ≤
12 ( k T k A + k T k A ) . (3.2) Proof.
Let T = (cid:18) O T T ♯ A O (cid:19) . In view of Lemma 3.2, we have T ∈ B A / ( H ⊕ H )as B A ( H ) ⊆ B A / ( H ). So, by Proposition 1.2 there exists e T ∈ B ( R ( A / )) suchthat Z A T = e T Z A . Moreover, by applying Lemma 3.2 together with Lemma 1.4,we get e T = O f T g T ♯ A O ! = O f T f T ∗ O ! So, by applying Lemma 1.3 together with Theorem 3.3 we infer that12 k f T + f T k B ( R ( A / )) ≤ ω ( e T ) ≤
12 ( k f T k B ( R ( A / )) + k f T k B ( R ( A / )) ) . On the other hand, it is not difficult to see that f T + f T = ^ T + T . Therefore,(3.2) is proved by using Lemma 1.3. (cid:3) Using the inequality (3.2) we prove the following theorem.
Theorem 3.5.
Let T , T ∈ B A ( H ) be such that T k A T . Let also A = (cid:18) A A (cid:19) . Then ω A (cid:18) O T e − iβ T ♯ O (cid:19) = ( k T k A + k T k A ) , for some real β .Proof. We have k T + e iβ T k A = k T k A + k T k A for some real β , as T k A T .Therefore, using Theorem 3.4 we have k T k A + k T k A = k T + e iβ T k A ≤ ω A (cid:18) O T e − iβ T ♯ O (cid:19) ≤ k T k A + k T k A . It follows that ω A (cid:18) O T e − iβ T ♯ O (cid:19) = ( k T k A + k T k A ) , for some real β . (cid:3) Using the notion of A -numerical radius orthogonality we obtain A -numericalradius inequality for d × d operator matrices in the following theorem. Theorem 3.6.
Let T = ( T ij ) be a d × d operator matrix where T ij ∈ B A / ( H ) .Then, ω A ( T ) ≥ max { ω A ( T ii ) , ω A ( S i ) : 1 ≤ i ≤ d } where for each i ∈ { , · · · , d } , the operator matrix S i = ( s ijk ) d × d is defined as s ijk = O if j = i or k = i and s ijk = T jk otherwise, i.e. S i = T . . . T i − O T i +1) . . . T d ... ... . . . ... ... ... . . .T ( i − . . . T ( i − i − O T ( i − i +1) . . . T ( i − d O . . . O O O . . . OT ( i +1)1 . . . T ( i +1)( i − O T ( i +1)( i +1) . . . T ( i +1) d ... . . . ... ... ... . . . ... T d . . . T d ( i − O T d ( i +1) . . . T dd . Proof.
First we prove that ω A ( T ) ≥ ω A ( T ii ) for all i ∈ { , · · · , d } . By definitionof the A -numerical radius of the operator T , there exists a sequence of A -unitvectors { x n } in H such thatlim n → + ∞ |h T x n | x n i A | = ω A ( T ) . Let X n = ( x n , , · · · , ∈ H . Then k X n k A = k x n k A = 1 . Therefore, from an easycalculation we have |h T X n , X n i A | = |h T x n | x n i A |⇒ ω A ( T ) ≥ |h T x n | x n i A |⇒ ω A ( T ) ≥ lim n → + ∞ |h T x n | x n i A |⇒ ω A ( T ) ≥ ω A ( T ) . Similarly, we can show that ω A ( T ) ≥ ω A ( T ii ) for all i = 2 , , . . . , d .Next we will prove that ω A ( T ) ≥ ω A ( S i ) for all i ∈ { , · · · , d } . Let usassume that M i = T − S i . We first show that S i ⊥ ω A M i . Let { X m } = { ( x m , x m , · · · , x md ) } be a sequence of A -unit vector in H , i.e. k X m k A = 1,i.e. k x m k A + k x m k A + · · · + k x md k A = 1 such thatlim m → + ∞ |h S i X m , X m i A | = ω A ( S i ) . We claim that there exist sequence { Z m } in H of the form { ( z m , z m , . . . , z m ( i − , ,z m ( i +1) , . . . , z md ) } such that k Z m k A = 1 andlim m → + ∞ |h S i Z m , Z m i A | = ω A ( S i ) . -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 15 Suppose that P dj =1 ,j = i k x mj k A <
1. Let α = qP dj =1 ,j = i k x mj k A and Z m = α ( x m , x m , . . . , x m ( i − , , x m ( i +1) , . . . , x md ) . Then k Z m k A = 1. Since α >
1, we have |h S i Z m , Z m i A | = α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X j =1 ,j = i n X k =1 ,k = i h T jk x mk | x mj i A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = α |h S i X m , X m i A | > |h S i X m , X m i A | . So, we infer that ω A ( S i ) ≥ lim m → + ∞ |h S i Z m , Z m i A | ≥ lim m → + ∞ |h S i X m , X m i A | = ω A ( S i ) . This implies that lim m → + ∞ |h S i Z m , Z m i A | = ω A ( S i ) . This proves our claim.Now, an easy calculation shows that h M i Z m , Z m i A = 0 . Therefore, for each β ∈ [0 , π ), ℜ e (cid:0) e iβ h Z m , S i Z m i A h M i Z m , Z m i A (cid:1) = 0 . Therefore from Theorem 2.5 we have S i ⊥ ω A M i . So, ω A ( T ) = ω A ( S i + 1 × M i ) ≥ ω A ( S i ) for all i = 1 , , . . . , d. Hence we conclude that ω A ( T ) ≥ max { ω A ( T ii ) , ω A ( S i ) : 1 ≤ i ≤ d } , This completes the proof of the theorem. (cid:3)
Based on Theorem 3.6 we obtain the following inequality.
Corollary 3.7.
Let T = ( T ij ) be a d × d operator matrix where T ij ∈ B A / ( H ) .Then, ω A ( T ) ≥ max (cid:26) ω A ( T kk ) , ω A ′ (cid:18) T ii T ij T ji T jj (cid:19) : 1 ≤ k ≤ d, ≤ i < j ≤ d (cid:27) where A ′ = (cid:18) A A (cid:19) .Proof. Let D = diag ( A, A, . . . , A ) be a ( d − × ( d −
1) operator matrix and for i ∈ { , · · · , d } we set the following ( d − × ( d −
1) operator matrix R i = T . . . T i − A i +1) . . . T d ... . . . ... ... . . . ... T ( i − . . . T ( i − i − T ( i − i +1) . . . T ( i − d T ( i +1)1 . . . T ( i +1)( i − T ( i +1)( i +1) . . . T ( i +1) d ... . . . ... ... . . . ... T d . . . T d ( i − T d ( i +1) . . . T dd . It can be seen that ω A ( S i ) = ω D ( R i ) where S i is defined as in Theorem 3.6.Moreover, by applying Theorem 3.6 repeatedly on R i for each i = 1 , , . . . , d weget our desired inequality. (cid:3) In the following lemma, we will show that A-numerical radius of semi-Hilberticanspace operators satisfying weak A-unitary invariance property.
Lemma 3.8.
Let T ∈ B A / ( H ) . Then, ω A ( U ♯ T U ) = ω A ( T ) , (3.3) for every A -unitary operator U ∈ B A ( H ) .Proof. Since T ∈ B A / ( H ) and U ∈ B A ( H ) ⊆ B A / ( H ), then by Proposition 1.2there exists a unique e T , e U ∈ B ( R ( A / )) such that Z A T = e T Z A and Z A U = e U Z A .On the other hand, since U ∈ B A ( H ) is an A -unitary operator, then( U ♯ A ) ♯ A U ♯ A = U ♯ A U = P A . This gives ^ U ♯ A U = ^ ( U ♯ A ) ♯ A U ♯ A = f P A . So, it can be seen by using Lemma 1.4 that e U ∗ e U = e U e U ∗ = I R ( A / ) . Hence, e U is an unitary operator on the Hilbert space R ( A / ). So, we get ω ( e U ∗ e T e U ) = ω ( e T ) . This implies, by Lemma 1.4 that ω ( f U ♯ e T e U ) = ω ( e T ) , which in turn yields that ω ( ^ U ♯ T U ) = ω ( e T ) , Therefore, we obtain (3.3) by using Lemma 1.3. (cid:3)
Next, we obtain a lower bound for A -numerical radius of 2 × Lemma 3.9.
Let T = (cid:18) T T T T (cid:19) where T ij ∈ B A / ( H ) and A = (cid:18) A A (cid:19) .Then ω A ( T ) = ω A (cid:18) T iT − iT T (cid:19) . Proof.
Let U = (cid:18) − iI OO I (cid:19) . By using Lemma 3.1, one gets U ♯ A U = (cid:18) iP A OO P A (cid:19) (cid:18) − iI OO I (cid:19) = (cid:18) P A OO P A (cid:19) = P A , where P A is denoted to be the orthogonal projection onto R ( A ). Similarly, weshow that ( U ♯ A ) ♯ A U ♯ A = P A . Hence, U is an A -unitary operator. So, the desired -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 17 equality follows immediately by using (3.3) and remarking that ω A ( T ) = ω A ( P A T ). (cid:3) Now we are in a position to prove the following lower bound for A -numericalradius of 2 × A = diag ( A, A ). Theorem 3.10.
Let T = (cid:18) T T T T (cid:19) where T ij ∈ B A / ( H ) and A = (cid:18) A A (cid:19) .Then ω A ( T ) ≥ max { ω A ( T ) , ω A ( T ) , α, β } where α = s m A (cid:18) T + T (cid:19) + ω A (cid:18) T + T (cid:19) ,β = s m A (cid:18) T + T (cid:19) + ω A (cid:18) T − T (cid:19) . Proof.
First we prove that ω A ( T ) ≥ ω A ( T ii ) for all i = 1 , . Let { x n } be a sequenceof A -unit vectors in H such thatlim n → + ∞ |h T x n | x n i A | = ω A ( T ) . Let X n = ( x n , ∈ H ⊕ H . Then k X n k A = k x n k A = 1 . Therefore, from an easycalculation we have |h T X n , X n i A | = |h T x n | x n i A |⇒ ω A ( T ) ≥ |h T x n | x n i A |⇒ ω A ( T ) ≥ lim n → + ∞ |h T x n | x n i A |⇒ ω A ( T ) ≥ ω A ( T ) . Similarly, we can show that ω A ( T ) ≥ ω A ( T ) . Now we show that ω A ( T ) ≥ α and ω A ( T ) ≥ β . We consider T = P + S where P = (cid:18) T OO T (cid:19) and S = (cid:18) O T T O (cid:19) . Let { z n } be a sequence of A -unitvectors in H such thatlim n → + ∞ (cid:12)(cid:12) h ( T + T ) z n | z n i A (cid:12)(cid:12) = ω A (cid:0) T + T (cid:1) . Let Z n = √ ( z n , z n ) and Z n = √ ( − z n , z n ) be in H ⊕ H . Then by an easycalculation we see that h S Z n , Z n i A = −h S Z n , Z n i A = h T + T z n | z n i A , h P Z n , Z n i A = h P Z n , Z n i A = h T + T z n | z n i A . From this we observe that either one of the following holds:( i ) ℜ e n h P Z n , Z n i A h S Z n , Z n i A o ≥ . ( ii ) ℜ e n h P Z n , Z n i A h S Z n , Z n i A o ≥ . Without loss of generality we assume that ( i ) holds. Then, we have ω A ( T ) = ω A ( P + S ) ≥ |h ( P + S ) Z n , Z n i A | = |h P Z n , Z n i A + h S Z n , Z n i A | = |h P Z n , Z n i A | + |h S Z n , Z n i A | + 2 ℜ e n h P Z n , Z n i A h S Z n , Z n i A o ≥ |h P Z n , Z n i A | + |h S Z n , Z n i A | = (cid:12)(cid:12) h (cid:0) T + T (cid:1) z n | z n i A (cid:12)(cid:12) + (cid:12)(cid:12) h (cid:0) T + T (cid:1) z n | z n i A (cid:12)(cid:12) ≥ m A (cid:0) T + T (cid:1) + (cid:12)(cid:12) h (cid:0) T + T (cid:1) z n | z n i A (cid:12)(cid:12) ≥ m A (cid:0) T + T (cid:1) + lim n → + ∞ (cid:12)(cid:12) h (cid:0) T + T (cid:1) z n | z n i A (cid:12)(cid:12) = m A (cid:0) T + T (cid:1) + ω A (cid:0) T + T (cid:1) ⇒ ω A ( T ) ≥ q m A (cid:0) T + T (cid:1) + ω A (cid:0) T + T (cid:1) . To show ω A ( T ) ≥ β , we consider the operator matrix (cid:18) T iT − iT T (cid:19) . By re-placing T , T by iT , − iT respectively in the above last inequality, and byusing Lemma 3.9 we have ω A ( T ) ≥ q m A (cid:0) T + T (cid:1) + ω A (cid:0) iT − iT (cid:1) = q m A (cid:0) T + T (cid:1) + ω A (cid:0) T − T (cid:1) . Therefore, we conclude that ω A ( T ) ≥ max { ω A ( T ) , ω A ( T ) , α, β } . Hence, completes the proof of the theorem. (cid:3)
Remark . Here we would like to remark that the acquired inequality in The-orem 3.10 generalized as well as improves on the inequality obtained in [16, Th.3.1].The following corollary is an immediate consequence of Corollary 3.7 and The-orem 3.10.
Corollary 3.12.
Let T = ( T ij ) be a d × d operator matrix where T ij ∈ B A / ( H ) .Then, ω A ( T ) ≥ max { ω A ( T kk ) , α ij , β ij : 1 ≤ k ≤ d, ≤ i < j ≤ d } where α ij = r m A (cid:16) T ii + T jj (cid:17) + ω A (cid:16) T ij + T ji (cid:17) ,β ij = r m A (cid:16) T ii + T jj (cid:17) + ω A (cid:16) T ij − T ji (cid:17) . -NUMERICAL RADIUS ORTHOGONALITY AND PARALLELISM 19 Remark . Here we remark that the inequality in Corollary 3.12 generalisedand improves on the existing inequality in [16, Th. 3.3]. The inequality in [16,Th. 3.3] follows from Corollary 3.12 by considering A = I. Our next result reads as follows.
Theorem 3.14.
Let T = ( T ij ) be a d × d operator matrix where T ij ∈ B A / ( H ) and T ij = O when i > j , i.e. T = T T T . . . T d O T T . . . T d ... . . . . . . . . . ...... . . . . . . T ( n − d O . . . . . . O T dd . Then, we have ω A ( T ) ≥ max n ω A ( T kk ) , k T ij k A : 1 ≤ k ≤ d, ≤ i < j ≤ d o . Proof.
We have from Corollary 3.7 that ω A ( T ) ≥ max (cid:26) ω A ( T kk ) , ω A ′ (cid:18) T ii T ij O T jj (cid:19) : 1 ≤ k ≤ d, ≤ i < j ≤ d (cid:27) where A ′ = (cid:18) A A (cid:19) . So, to prove this theorem we only show that ω A ′ (cid:18) T ii T ij O T jj (cid:19) ≥ k T ij k A . Let us assume that P = (cid:18) O T ij O O (cid:19) and S = (cid:18) T ii OO T jj (cid:19) . It is not difficult toverify that k P k A ′ = k T ij k A . Moreover, since A ′ P = O , then by [14, Cor. 2.2]we have ω A ′ ( P ) = k P k A ′ = k T ij k A . We claim that P ⊥ ω A ′ S . In view of (1.3), thereexists a sequence of A -unit vectors { x m } in H such thatlim m → + ∞ k T ij x m k A = k T ij k A . Let Z m = √ k T ij k A ( T ij x m , k T ij k A x m ) and Z m = √ k T ij k A ( − T ij x m , k T ij k A x m ) bein H ⊕H . Then it can be checked that, for the sequences of A ′ -unit vectors { Z m } and { Z m } , lim m → + ∞ h P Z m , Z m i A ′ = k T ij k A , i.e. lim m → + ∞ |h P Z m , Z m i A ′ | = ω A ′ ( P )lim m → + ∞ h P Z m , Z m i A ′ = −k T ij k A , i.e. lim m → + ∞ |h P Z m , Z m i A ′ | = ω A ′ ( P )Also we have h S Z m , Z m i A ′ = h S Z m , Z m i A ′ . Therefore for any β ∈ [0 , π ), eitherone of the following holds:( i ) ℜ e (cid:8) e iβ h Z m , P Z m i A ′ h S Z m , Z m i A ′ (cid:9) ≥ . ( ii ) ℜ e (cid:8) e iβ h Z m , P Z m i A ′ h S Z m , Z m i A ′ (cid:9) ≥ . Therefore from Theorem 2.5, we have P ⊥ ω A ′ S . So, ω A ′ (cid:18) T ii T ij O T jj (cid:19) = ω A ′ ( P + 1 × S ) ≥ ω A ′ ( P ) = k T ij k A . Hence, completes theproof. (cid:3)
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E-mail address : [email protected] ; [email protected] [2] University of Sfax, Tunisia.
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