Numerical Radius and Operator Norm Inequalities

For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$.

Let denote the C* –algebra of all bounded linear operators on a Hilbert space . Among other results, we show that if are nonzero, then where denote the operator norm and the numerical radius, respectively. This provides a significant improvement and reverse of the known bound To achieve this, we treat the triangle inequality and a well-known identity for the numerical radius of an operator matrix. Using the same technique, we then present an inequality for the spectral radius of the sum of two operators. The results obtained are compared with those of the existing literature. Many other results are shown as refinements or reverses of other known facts about the numerical radius and the usual operator norm.

We develop several $\ell ^p$ -operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$ -sum of Banach spaces. Using these inequalities, we obtain p -numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p -numerical radius bound for the Kronecker product $A\otimes B$ , where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$ . This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$ -operator norm and p -numerical radius of $A\in {M}_k(\mathbb {C})$ , respectively, then it is shown that $$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$ where $\mu _p(A)$ is a positive real number that involves the $\ell ^p$ -operator norms of the Cartesian decomposition of A . In addition, a complete characterization of the equality case $\frac {1}{2}\|A\|_p= w_p(A)$ is given.

Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set
\n W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1}
\n
\nWriting T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set 
\n{(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}.
\n
\nThis leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by 
\nW(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}.
\nThe joint numerical range has been studied extensively in order to understand the
\njoint behaviour of operators.
\n
\nLet K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by 
\nW_(e ) (T)=∩{W(T+K) :K∈K(H) }.
\n
\nThe joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by 
\nW_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }.
\nThese notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given.
\n
\nIn 2010, Müller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979.
\n
\nLet H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. 
\nThe maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range.
\nHowever the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong.

In this paper, we prove some singular value inequalities for sum and product of operators. Also, we obtain several generalizations of recent inequalities. Moreover, as applications we establish some unitarily invariant norm and trace inequalities for operators which provide refinements of previous results.

We introduce a new norm on the space of all bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis–Wielandt radius norm. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate that our results improve on some of the important existing numerical radius inequalities. Other application of this new norm have also studied.

We show that a bounded linear operator is a multiple of a unitary operator if and only if and always have the same numerical radius or the same numerical range for all (rank one) . More generally, for any bounded linear operators , we show that and always have the same numerical radius (resp., the same numerical range) for all (rank one) if and only if (resp., ) is a multiple of a unitary operator for some . We extend the result to other types of generalized numerical ranges including the -numerical range and the higher rank numerical range.

Better bounds on the numerical radii of Hilbert space operators

‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

We present several upper bounds for the numerical radii of 2 × 2 operator matrices. We employ these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if T is a bounded linear operator on a complex Hilbert space, then for every r ≥ 1 and α ∈ [0, 1]. This substantially improves on the existing inequality . Here w(.) and ||.|| denote the numerical radius and the usual operator norm, respectively.

We present a collection upper bounds for the numerical radii of a certain 2 × 2 operator matrices. We use these bounds to improve on some known numerical radius inequalities for powers of Hilbert space operators. In particular, we show that if 𝐴 is a bounded linear operator on a complex Hilbert space, then 𝑤 2𝑟 (𝐴) ≤ 1+𝛼 8 ‖|𝐴| 2𝑟 +|𝐴 ∗ | 2𝑟‖+ 1+𝛼 4 𝑤(|𝐴| 𝑟 |𝐴 ∗ | 𝑟 )+ 1−𝛼 2 𝑤 𝑟 (𝐴 2 ) for every r ≥ 1 and α ∈ [0,1]. This substantially improves on the existing inequality 𝑤 2𝑟 (𝐴) ≤ 1 2 ‖|𝐴| 2𝑟 + |𝐴 ∗ | 2𝑟‖. Here 𝑤(. ) and ||. || denote the numerical radius and the usual operator norm, respectively.

Let A, B, X, and Y be bounded linear operators on a complex Hilbert space. It is shown that where w(·) and ‖·‖ are the numerical radius and the usual operator norm, respectively. This inequality includes and improves upon earlier numerical radius inequalities proved in this context. Applications of this inequality are given to obtain new numerical radius inequalities for commutators of self-adjoint and positive operators.

Spectral radius, numerical radius, and the product of operators

Let V be the C∗-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i1, . . . , im) with i1, . . . , im ∈ {1, . . . , k}, define a product of A1, . . . , Ak ∈ V by A1 ∗ · · · ∗ Ak = Ai1 . . . Aim . This includes the usual product A1 ∗ · · · ∗ Ak = A1 · · ·Ak and the Jordan triple product A ∗ B = ABA as special cases. Denote the numerical range of A ∈ V by W (A) = {(Ax, x) : x ∈ H, (x, x) = 1}. If there is a unitary operator U and a scalar μ satisfying μ = 1 such that φ : V→ V has the form A 7→ μU∗AU or A 7→ μU∗AtU, then φ is surjective and satisfies W (A1 ∗ · · · ∗ Ak) =W (φ(A1) ∗ · · · ∗ φ(Ak)) for all A1, . . . , Ak ∈ V. It is shown that the converse is true under the assumption that one of the terms in (i1, . . . , im) is different from all other terms. In the finite dimensional case, the converse can be proved without the surjective assumption on φ. An example is given to show that the assumption on (i1, . . . , im) is necessary. 2000 Mathematics Subject Classification. 47A12, 47B15, 47B49, 15A60, 15A04, 15A18

A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.