Some numerical radius inequalities for semi-Hilbert space operators Kais Feki1 Abstract. Let A be a positive bounded linear operator acting on a complex Hilbert space , . Let ωA(T ) and T A denote the A-numerical radius and the A-operatorH h· |seminorm ·i of an operatork k T acting on the semi-Hilbertian space , A respectively, where x y A := Ax y for all x, y . In this paper,H h· we | ·i show that h | i h | i ∈ H 1 ♯A ♯A 2 1 ♯A ♯A T T + TT A ω (T ) T T + TT A. 4 k k ≤ A ≤ 2 k k Here T ♯A is denoted to be a distinguished A-adjoint operator of T . Moreover, a considerable improvement of the above inequalities is proved. This allows I T to compute the A-numerical radius of the operator matrix where 0 I − A = diag(A, A). In addition, several A-numerical radius inequalities for semi- Hilbertian space operators are also established. 1. Introduction and Preliminaries In this paper, let , be a complex Hilbert space equipped with the H h· | ·i norm . Let ( ) stand for the C∗-algebra of all bounded linear operators definedk·k on withB theH identity operator I. An operator T ( ) is said to be positive if HT x x 0 for all x . The square root of a positive∈ B H operator T is h 1/|2 i ≥ ∈ H denoted by T . Let T denotes the square root of T ∗T , where T ∗ is the adjoint of T . The operator norm| | and the spectral radius of an operator T are denoted by T and r (T ) respectively and they are given by k k T = sup T x ; x , x =1 . k k {k k ∈ H k k } arXiv:2001.00398v2 [math.FA] 17 Apr 2020 and r(T ) = sup λ ; λ σ(T ) , {| | ∈ } where σ(T ) denotes the spectrum of T . It is well-known that, for every T ( ), we have ∈ B H 2 2 2 2 2 T = T = T ∗ = r( T )= r( T ∗ ). (1.1) k k | | | | | | | | Date: April 20, 2020. 2010 Mathematics Subject Classification. Primary 47A12, 46C05; Secondary 47B65, 47A05. Key words and phrases. Positive operator, semi-inner product, numerical radius, A-adjoint operator, inequality. [1] University of Sfax, Sfax, Tunisia. E-mail address: [email protected] 1 2 Kais Feki From now on, A always stands for a positive linear operator in ( ). The cone of all positive operators of ( ) will be denoted by ( )+. InB thisH article, for a given A ( )+, we areB goingH to consider an additionalB H semi-inner product on ∈defined B H by h· | ·iA H x y = Ax y , x, y . h | iA h | i ∀ ∈ H This makes into a semi-Hilbertian space. The seminorm induced by is H h· | ·iA given by x A = x x A for every x . It can be observed that ( , A) is a normedk spacek if andh | onlyi if A is one-to-one∈ H and that the semi-HilbertianH k·k space p ( , A) is complete if and only if the range of A is closed in . In what follows, byH ank·k operator we mean a bounded linear operator in ( ).H Also, the range of every operator T is denoted by (T ), its null space by B H(T ). Given a linear R N M subspace of , denotes the closure with respect to the norm topology of . If is a closedH subspaceM of , then P stands for the orthogonal projectionH onto M. H M M The semi-inner product A induces on the quotient / (A) an inner prod- uct which is not completeh· unless | ·i (A) is closed. However,H aN canonical construc- tion due to de Branges and RovnyakR [12] shows that the completion of / (A) is isometrically isomorphic to the Hilbert space (A1/2) equipped with theH followingN inner product R 1/2 1/2 1 2 A x, A y R(A / ) := P (A)x P (A)y , x, y . h i h R | R i ∀ ∈ H 1/2 1/2 For the sequel, the Hilbert space (A ), , R 1/2 will be denoted by R(A ). R h· ·i (A ) Definition 1.1. ([3]) Let T ( ). An operatorS ( ) is called an A- adjoint of T if for every x, y ∈ B, theH identity T x y ∈= BxHSy holds. That ∈ H h | iA h | iA is AS = T ∗A. Generally, the existence of an A-adjoint operator is not guaranteed. The set of all operators in ( ) admitting A-adjoints is denoted by A( ). By Douglas Theorem [15], we haveB H B H ( )= T ( ); (T ∗A) (A) . BA H { ∈ B H R ⊆ R } 1/2 Further, the set of all operators admitting A -adjoints is denoted by A1/2 ( ). Again, by applying Douglas Theorem, we obtain B H 1/2 ( )= T ( ); λ> 0 ; T x λ x , x . BA H { ∈ B H ∃ k kA ≤ k kA ∀ ∈ H} If T 1/2 ( ), we will say that T is A-bounded. Notice that ( ) and ∈ BA H BA H 1/2 ( ) are two subalgebras of ( ) which are, in general, neither closed nor BA H B H dense in ( ). Moreover, the following inclusions A( ) A1/2 ( ) ( ) hold withB equalityH if A is injective and has closedB range.H ⊆ For B anH account⊆ B H of results, we refer to [3, 4, 8, 16] and the references therein. Clearly, induces h· | ·iA a seminorm on 1/2 ( ). Indeed, if T 1/2 ( ), then BA H ∈ BA H T x A T A := sup k k = sup T x A ; x , x A =1 < . k k x (A), x A k k ∈ H k k ∞ ∈xR=0 k k 6 Some numerical radius inequalities for semi-Hilbert space operators 3 Notice that, if T 1/2 ( ), then T = 0 if and only if AT = 0. Further, it ∈ BA H k kA was proved in [20] that for T 1/2 ( ) we have ∈ BA H T = sup T x y ; x, y , x = y =1 . (1.2) k kA {|h | iA| ∈ H k kA k kA } It should be emphasized that it may happen that T = + for some T k kA ∞ ∈ ( ) 1/2 ( ) (see [16, Example 2]). B H \ BA H Before we move on, it should be mentioned that for T 1/2 ( ) we have ∈ BA H T x T x , x . (1.3) k kA ≤k kAk kA ∀ ∈ H Also, we would like to emphasize that (1.3) fails to hold in general for some 0 0 0 1 T ( ). In fact, one can take the operators A = and T = on ∈ B H 0 1 1 0 2 C . If x = (1, 0), then x A = 0 and T x A = 1. Thus, (1.3) fails to be true. Moreover, by applying (k1.3k) we show thatk k TS T S , (1.4) k kA ≤k kAk kA for every T,S 1/2 ( ). ∈ BA H If T A( ), then by Douglas theorem there exists a unique A-adjoint of T , ∈ B H♯ ♯ ♯ denoted by T A , which satisfies (T A ) (A). We observe that T A = A†T ∗A, R ⊆ R ♯ where A† is the Moore-Penrose inverse of A. For results concerning T A and A† see ♯A ♯A ♯A [3, 4]. Notice that if T A( ), then T A( ) and (T ) = P (A)TP (A). ∈ B H ∈ B H R R For more facts related to this class of operators, we invite the reader to [3, 4] and their references. Now, we recall that an operator T ( ) is said to be A-isometry if T x = x for all x . Further, an operator∈ B H U ( ) is k kA k kA ∈ H ∈ BA H called A-unitary if U and U ♯A are A-isometries. By an A-normal operator, we ♯A ♯A mean an operator T A( ) which satisfies T T = T T . For more details related to these classes∈ of B operators,H the reader can consult [3, 9]. Any operator T ( ) can be represented as T = (T )+ i (T ), where ∈ BA H ℜA ℑA T + T ♯A T T ♯A (T ) := and (T ) := − . ℜA 2 ℑA 2i Recently, the A-numerical range of an operator T ( ) is defined by Baklouti ∈ B H et al. in [8] as WA(T ) = T x x A ; x , x A = 1 . It was shown in [8] that W (T ) is a nonemptyh convex| i subset∈ of H Ck whichk is not necessarily closed A even if dim( ) < . Notice that supremum modulus of WA(T ) is called the A-numericalH radius∞ of T (see [8]). More precisely, we have ω (T ) = sup λ ; λ W (T ) = sup T x x ; x , x =1 . A | | ∈ A h | iA ∈ H k kA It should be mentioned that W (T )= C when T ( ) and satisfies T ( ( A)) A (A) ([8, Theorem 2.1.]). So, ω (T )=+ for∈ every B H operator T N( ) such6⊆ N A ∞ ∈ B H that T ( (A)) (A). Notice that ωA( ) is a seminorm on A1/2 ( ) with is equivalentN to the6⊆A N-operator seminorm. More· precisely, it was shownB H in [8] that for every T 1/2 ( ), we have ∈ BA H 1 T ω (T ) T . (1.5) 2 k kA ≤ A ≤k kA 4 Kais Feki By an A-selfadjoint, we mean an T ( ) which satisfies AT is selfadjoint, ∈ B H that is, AT = T ∗A. Further, for every A-selfadjoint operator T we have T = ω (T ) := sup T x x ; x , x =1 . (1.6) k kA A {|h | iA| ∈ H k kA } (see [16]).