Some generalizations of numerical radii inequalities

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.

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 generalize several inequalities involving powers of the numerical radius for the product of two operators acting on a Hilbert space. Moreover, we give a Jensen operator inequality for strongly convex functions. As a corollary, we improve the operator Holder-McCarthy inequality under suitable conditions. In particular, we prove that if $f:Jrightarrow mathbb{R}$ is strongly convex with modulus $c$ and differentiable on ${rm int}(J)$ whose derivative is continuous on ${rm int}(J)$ and if $T$ is a self-adjoint operator on the Hilbert space $cal{H}$ with $sigma(T)subset {rm int}(J)$, then $$langle T^2x,xrangle-langle Tx,xrangle^2leq dfrac{1}{2c}(langle f'(T)Tx,xrangle -langle Tx,xrangle langle f'(T)x,xrangle)$$ for each $xincal{H}$, with $|x|=1$.

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.

The numerical range W(T) of a linear bounded operator T on a Hilbert space is the collection of complex numbers of the form (Tx,x) with x ranging over the unit vectors in the Hilbert space. In this study, firstly, the connection between numerical range of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the first investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical range of the operator A is in the form W(A)= co (⋃_{n=1}^{∞}W(A_{n}) ), where co (Ω), Ω⊂C denotes the convex hull of Ω. The numerical radius w(T) of a linear bounded operator T on a Hilbert space is a number which is given by w(T)=sup {|λ| : λ∈W(T) }. In this study, secondly, the connection between numerical radius of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the second investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical radius of the operator A is in the following form w(A)= sup w(A_{n}). The Crawford number c(T) of a linear bounded operator T on a Hilbert space is given by c(T)=inf {|λ| : λ∈W(T) }. In this study, thirdly, the connection between Crawford number of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the third investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), Re(A_{n})≥0 (≤0), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then Crawford number of the operator A is in the following form c(A)=inf c(A_{n}). Moreover, It is known that for a linear bounded accretive operator T if W(T)⊂{z ∈C:|argz| < ϕ} for any ϕ∈[0, π/ 2), then it is called a sectorial operator with vertex γ=0 and semi-angle ϕ. In this case, T ∈S_{ϕ}(H). In this study, finally, the connection between sectoriality of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the last investigation the result obtained is as follows: If for any n≥1,H_{n} is a Hilbert space, A_{n}∈L(H_{n}),A_{n}∈S_{ϕ_{n}}(H_{n}),H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n},A∈L(H), then for some ϕ∈[0,π/2), A∈S_{ϕ}(H) the necessary and sufficient condition is sup ϕ_{n}<ϕ. At the end of this study, an example including all these connections has been given.

This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .

Abstract. Let A and B be two unital C ∗ -algebras. Denote by W(a)the numerical range of an element a∈ A. We show that the conditionW(ax) = W(bx),∀x∈ A implies that a= b. Using this, among otherresults, it is proved that if φ : A → B is a surjective map such thatW(φ(a)φ(b)φ(c)) = W(abc) for all a,band c∈A, then φ(1) ∈Z(B) andthe map ψ= φ(1) 2 φis multiplicative. 1. IntroductionLet A be a C ∗ -algebra with unit 1 and let S(A) be the state space of A, i.e.,S(A) = {ϕ∈ A ′ : ϕ≥ 0,ϕ(1) = 1} (here A ′ is the topological dual of A). Foreach a∈ A, the algebraic numerical range V(a) and numerical radius v(a) aredefined byV(a) = {f(a) : f∈ S(A)} and v(a) = sup z∈V (a) |z|.By the Gelefand-Naimark theorem, every C ∗ -algebra may be viewed as a closed∗-subalgebra of B(H) where B(H) denotes the algebra of all bounded linearoperators acting on a Hilbert space H. It is well known that V(a) is the closureof W(a) and v(a) = w(a) = sup λ∈W(a) |λ|, where W(a) = {(at,t) : t∈ H,ktk = 1}and (,) denotes the inner product. Here W(a) is called the usual numericalrange of the operator a.In the last few decades, there has been a considerable interest in the problemof characterization of maps that preserves the numerical range or the numericalradius, see for instance the papers [4, 12, 13, 15] and the references therein.Notice that, based on the aforesaid, preserving the usual numerical range Wimplies the preservation of the spacial numerical range V. Therefore, we willconcentrate our study henceforth on W. Recently, Hou and Di described in [9]surjective maps on the algebra B(H) which preserves the numerical range of

In this note we report on the recently introduced concept of the numerical range of a bounded linear operator on a Hilbert space with respect to a family of projections. The importance of this new concept lies in the fact that it unifies and generalizes well‐established versions of the numerical range such as the classical numerical range introduced by Toeplitz and Hausdorff, the quadratic numerical range as well as the block numerical range. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A general inequality involving powers of the numerical radius for sums and products of Hilbert space operators is given. This inequality generalizes several recent inequalities for the numerical radius, and includes that if and are operators on a complex Hilbert space , then for . It is also shown that if is normal , then . Related numerical radius and usual operator norm inequalities for sums and products of operators are also presented.

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Inner functions of numerical contractions

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

In this paper we give upper and lower bounds of the infimum of k&amp;nbsp; such that kI+2ReT&amp;otimes;Sm&amp;nbsp; is positive, where Sm&amp;nbsp; is the m&amp;times;m&amp;nbsp; matrix whose entries are all 0&amp;rsquo;s except on the superdiagonal where they are all 1&amp;rsquo;s and T&amp;isin;BH&amp;nbsp; for some Hilbert space H. &#x0D; &#x0D; When T&amp;nbsp; is self-adjoint, we have the minimum of k. &#x0D; &#x0D; When m=3&amp;nbsp; and T&amp;isin;B(H)&amp;nbsp; , we obtain the minimum of k&amp;nbsp; and an inequality&#x0D; &#x0D; Involving the numerical radius w(T) .

It is shown that $iB$ generates a strongly continuous group of exponential type $\omega$ on a Hilbert space if and only if for all $\alpha > \omega$, $B$ is similar to an operator with spectrum and numerical range contained in the horizontal strip $\{z \in {\bf C} \, | \, |Im(z)| < \alpha \}$.

We give new necessary and sufficient conditions for the numerical range W(T) of an operator T∈B(H) to be a subset of the closed elliptical set Kδ⊆C given byKδ=def{x+iy:x2(1+δ)2+y2(1−δ)2≤1}, where 0<δ<1. Here B(H) denotes the collection of bounded linear operators on a Hilbert space H. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. Specifically, we show that, if T acts on a finite-dimensional Hilbert space H and satisfies a certain genericity assumption, then W(T)⊆Kδ if and only if there exists a Hilbert space K⊇H, operators X1 and X2 on H and a unitary operator U acting on K such that(0.1)X1+X2=T,X1X2=δ and(0.2)X1k+X2k=2PHUk|H,k=1,2,…, where PH denotes the orthogonal projection from K to H.We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators T that satisfy the relations (0.1) and (0.2). This generalization yields a characterization of the operators T∈B(H) such that W(T) is contained in Kδ in terms of certain structured contractions that act on H⊕H.As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those T such that W(T)⊆Kδ. We prove that, if T acts on a finite-dimensional Hilbert space H, then W(T)⊆Kδ if and only if there exist a pair of contractions A,B∈B(H) such that A is self-adjoint andT=2δA+(1−δ)1+AB1−A. We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal H∞-extension problem for analytic functions defined on a slice of the symmetrized bidisc.