Upper bounds for the numerical radius of Hilbert space operators
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Upper bounds for the numerical radius of Hilbert space operators Akram Mansoori1 · Mohsen Erfanian Omidvar1 · Khalid Shebrawi2 Received: 6 July 2020 / Accepted: 25 September 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract New upper bounds for the numerical radius of Hilbert space operators are given. Moreover, we give some applications of our result in estimation of spectral radius. We also compare our results with some known results. Keywords Numerical radius · Norm inequality · Spectral radius Mathematics Subject Classification Primary 47A12 · 47A30 · Secondary 47B15
1 Introduction and preliminaries Let B(H) denote the C∗-algebra of all bounded linear operators on a complex Hilbert space H with inner product ⟨⋅, ⋅⟩ . For A ∈ B(H) , let
‖A‖ = sup{‖Ax‖ ∶ ‖x‖ = 1}, w(A) = sup{�⟨Ax, x⟩� ∶ ‖x‖ = 1}, r(A) = sup{�𝜆� ∶ 𝜆 ∈ 𝜎(A)}, denote the usual operator norm, the numerical radius and the spectral radius of A, respectively. It is well known that w(⋅) defines a norm on B(H) . This norm is equivalent to the operator norm. In fact, the following more precise result holds
1 ‖A‖ ≤ w(A) ≤ ‖A‖, 2
(1.1)
* Mohsen Erfanian Omidvar [email protected] Akram Mansoori [email protected] Khalid Shebrawi [email protected] 1
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
2
Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan
13
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r(A) ≤ w(A) ≤ ‖A‖.
(1.2)
The inequality (1.1) has been improved considerably by Kittaneh in [5] and [7]. It shows that if A ∈ B(H) , then
w(A) ≤
1 1 1 ‖(�A� + �A∗ �)‖ ≤ (‖A‖ + ‖A2 ‖ 2 ) ≤ ‖A‖, 2 2
(1.3)
and
1 ∗ 1 ‖A A + AA∗ ‖ ≤ w2 (A) ≤ ‖A∗ A + AA∗ ‖. 4 2
(1.4)
Considerable generalizations of the first inequality in (1.3) and the second inequality in (1.4) have been established in [9] for the numerical radius of one operator and for the sum of two operators. A general numerical radius inequality was proved by Kittaneh. It shows in [7] that if A, B, C, D, S, T ∈ B(H) , then
w(ATB + CSD) ≤
‖ 1‖ ‖A|T ∗ |2(1−𝛼) A∗ + B∗ |T|2𝛼 B + C|S∗ |2(1−𝛼) C∗ + D∗ |S|2𝛼 D‖ (1.5) ‖ 2‖ ‖ ‖
In [3], it has been shown that if A ∈ B(H) , then
wr (A) ≤
1 ‖ 2r𝛼 ‖ ‖|A| + |A∗ |2r(1−𝛼) ‖, ‖ 2‖
(1.6)
for r ≥ 1 and 𝛼 ∈ (0, 1) . In [9], Shebrawi and Albadawi extended the inequality (1.5). Let Ai,Bi,Xi ∈ B(H) (i = 1, 2, ..., n) , and let f and g be non-negative functions on [0, ∞) which are continuous and satisfy the relation f (t)g(t) = t for all t ∈ [0, ∞) . Then ( n ) n ‖ ∑ nr−1 ‖ ‖∑([ ∗ 2 (| ∗ |) ]r [ ∗ 2 (| |) ]r )‖ r ∗ Ai g |Xi | Ai + Bi f |Xi | Bi ‖ w Ai Xi Bi ≤ ‖ (1.7) ‖ 2 ‖ i=1 ‖ ‖ i=1 for all r ≥ 1 . In special cases, ( n ) n (( )r )‖ )r ( ∑ nr−1 ‖ 2𝛼 2(1−𝛼) ‖ ‖∑ r ∗ Ai + B∗i ||Xi || Bi A∗i ||Xi∗ || w Ai Xi Bi ≤ ‖, ‖ ‖ ‖ 2 i=1 ‖ ‖ i=1
(1.8)
and
( n ) n ∑ ( ))‖ nr−1 ‖ ‖ ‖∑( 2r (| |) f |Xi | + g2r ||Xi∗ || ‖ w Xi ≤ ‖ ‖ 2 ‖ i=1 i=1 ‖ ‖ r
(1.9)
for all r ≥ 1 and 0 ≤ 𝛼 ≤ 1. In this paper we introduce some refinements of numerical radius inequalities for Hilbert space ope
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