Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

Criterion for the Coincidence of Strong and Weak Orlicz Spaces

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We also find the exact value of the embedding constant which appears in the corresponding norm inequality.

On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$ ‖ a 2 ‖ ≤ ‖ a 2 + b 2 ‖ for $$a,b\in {\mathcal {A}}$$ a , b ∈ A is sufficient for a commutative real Banach algebra $${\mathcal {A}}$$ A with a unit to be isomorphic to the space $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) of continuous real-valued functions on a compact Hausdorff space $${\mathcal {K}}$$ K . Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of $${\mathcal {A}}$$ A onto $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) which are $$(1+\epsilon )$$ ( 1 + ϵ ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras $${\mathcal {A}}$$ A where $$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$ ‖ a 2 ‖ ≤ k ‖ a 2 + b 2 ‖ for all $$a,b\in {\mathcal {A}}$$ a , b ∈ A , for some $$k>1$$ k > 1 , but the inequality fails for $$k=1$$ k = 1 .

Some norm inequalities for upper sector matrices

We generalize some norm inequalities for 2 x 2 block accretive-dissipative matrices and positive semi-definite matrices that compare the diagonal blocks with the off-diagonal blocks. Moreover, we partially extend a norm inequality of n x n block accretive-dissipative matrices.

VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

Weighted norm inequality for bilinear Calderón–Zygmund operators on Herz–Morrey spaces with variable exponents

In this paper, we obtain a weighted norm inequality of bilinear Calderón–Zygmund operators in Herz–Morrey spaces with variable exponents and weight in the variable Muckenhoupt class.

L^p-norm inequality using q-moment and its applications

For a measurable function on a set which has a finite measure, an inequality holds between two Lp-norms. In this paper, we show similar inequalities for the Euclidean space and the Lebesgue measure by using a q-moment which is a moment of an escort distribution. As applications of these inequalities, we first derive upper bounds for the Renyi and the Tsallis entropies with given q-moment and derive an inequality between two Renyi entropies. Second, we derive an upper bound for the probability of a subset in the Euclidean space with given Lp-norm on the same set.