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Abstract

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely:

(va+(1v)b)2vj=1N2j(bab2j12j)2(avb1v)2+v2(ab)2

for v ∉ [0, 12N+1], N ∈ ℕ, a, b > 0; and

(va+(1v)b)2(1v)j=1N2j(aa2j1b2j)2(avb1v)2+(1v)2(ab)2

for v ∉ [1 − 12N+1, 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

Abstract

In this note, we introduce the asymptotic subspace confinement problem, generalizing the usual concept of convergence in discrete-time linear systems. Instead of precise convergence, subspace confinement only requires the convergence of states to a certain subspace of the state space, offering useful flexibility and applicability. We establish a criterion for deciding the asymptotic subspace confinement, drawing upon a general finiteness result for the infinite product of matrices. Our results indicate that the asymptotic subspace confinement problem is algorithmically decidable when an invariant subspace for the set of matrices and some polytope norms are given.

Abstract

We give an alternative lower bound for the numerical radii of Hilbert space operators. As a by-product, we find conditions such that

ω0RS0=R+S2

where R, S ∈ 𝔹(𝓗).

Abstract

In this paper, we prove that if a, b > 0 and 0 ≤ α ≤ 1, then for m = 1, 2, 3, . . . ,

r0m(am2-bm2)2r0m(bm+1-am+1b-a-(m+1)(ab)m2)(αa+(1-α)b)m-(aαb1-α)m,

where r 0 = min{α, 1 – α }. This is a considerable new generalization of two refinements of the Young inequality due to Kittaneh and Manasrah, and Hirzallah and Kittaneh, which correspond to the cases m = 1 and m = 2, respectively. As applications we give some refined Young type inequalities for generalized euclidean operator radius and the numerical radius of some well-know f -connection of operators and refined some Young type inequalities for the traces, determinants, and norms of positive definite matrices.

Abstract

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present

wpp(A1*T1B1,,An*TnBn)n1-1r21ri=1n[Bi*f2(|Ti|)Bi]rp+[Ai*g2(|Ti*|)Ai]rp1r-infx=1η(x),

where Ti,Ai,Bi𝔹() (1in), f and g are nonnegative continuous functions on [0,) satisfying f(t)g(t)=t for all t[0,), p,r1, N, and

η(x)=12i=1nj=1N((Ai*g2(|Ti*|)Ai)px,x2j-1-kj(Bi*f2(|Ti|)Bi)px,xkj2j
-(Bi*f2(|Ti|)Bi)px,xkj+1(Ai*g2(|Ti*|)Ai)px,x2j-1-kj-12j)2.

Abstract

We extend some numerical radius inequalities for adjointable operators on Hilbert C*-modules. A new refinement of a numerical radius inequality for some Hilbert space operators is given. More precisely, we prove that if T() is an invertible operator, then

T2T2+1T-122ω(T).

Abstract

In this paper by using the notion of sesquilinear form we introduce a new class of numerical range and numerical radius in normed space 𝒱, also its various characterizations are given. We apply our results to get some inequalities.

Abstract

The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where kλ=kλkλ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλΩ|A˜(λ)|=supλΩ|Akλ,kλ|. In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then

ber(AX±XA)ber12(A*A+AA*)ber12(X*X+XX*)

and

ber2(A*XB)X2ber(A*A)ber(B*B).

We also prove the multiplicative inequality

ber(AB)ber(A)ber(B)

Abstract

In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann–Liouville and Weyl.

Abstract

We illustrate a Bellman function technique in finding the modulus of uniform convexity of Lp spaces.