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In this article, we establish several Berezin number inequalities for the quadratic weighted operator geometric mean of operators on reproducing kernel Hilbert spaces and numerical radius inequalities for operators having similar positive parts. We also provide a concluding section which revisits the connection between the numerical radius and generalized inverses, and may spark new problems for future research interest.

In this paper, we first give two new upper bounds for the numerical radius of the product of two Hilbert space operators. The obtained bounds are compared numerically with previously known bounds. After that, the Hilbert–Schmidt numerical radius is studied for the generalized Aluthge transform and a pair of commuting operators. In the end, off-diagonal, tridiagonal, and anti-tridiagonal operator matrices are treated, where many upper bounds that extend some existing results are shown.

Let 𝔸=(AOOA)\mathbb{A}={\bigl{(}\begin{smallmatrix}A&O\\ O&A\\ \end{smallmatrix}\bigr{)}}be a 2×22\times 2diagonal operator matrix whose each diagonal entry is a bounded positive (semi-definite) linear operator Aacting on a complex Hilbert space ℋ\mathcal{H}. In this paper, we derive several 𝔸\mathbb{A}-numerical radius inequalities for 2×22\times 2operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator Aon ℋ\mathcal{H}. Some applications of our inequalities are also given.

Let Abe the 2×2diagonal operator matrix determined by a positive Hilbert space operator A. We give several upper bounds for the A-Berezin number of 2×2block matrices on a reproducing kernel Hilbert space and prove inequalities for the A-Berezin number of Hilbert space operators. Our results in this paper generalize and refine earlier the A-Berezin number inequalities.

This article presents some refinements and generalizations of existing numerical radius inequalities for Hilbert space operators. Further, several upper bounds for numerical radius with the Cartesian decomposition of an operator are provided. Also, we prove various upper bounds for the numerical radius of diagonal and off-diagonal operator matrices. Additionally, we establish certain numerical radius inequalities for operators using quadratic weighted operator geometric mean.

Diabetes is an important risk factor for atherosclerosis. The diabetic foot is characterized by the presence of arteriopathy and neuropathy. The vascular damage includes non-occlusive microangiopathy and macroangiopathy. Diabetic foot wounds are responsible for 5–10% of the cases of major or minor amputations. In fact, the risk of amputation of the lower limbs is 15–20% higher in diabetic populations than in the general population. The University of Texas classification is the reference classification for diabetic wounds. It distinguishes non-ischemic wounds from ischemic wounds which are associated with a higher rate of amputation. The first principles of treatment are the control of pain of an eventual infection. When ischemia is diagnosed, restoration of pulsatile blood flow by revascularization may be considered for salvaging the limb. The treatment options are angioplasty with or without stenting and surgical bypass or hybrid procedures combining the two. Distal reconstructions with anastomosis to the leg or pedal arteries have satisfactory limb-salvage rates. Subintimal angioplasty is a more recent endovascular technique. It could be suggested for elderly patients who are believed to be unsuitable candidates for a conventional bypass or angioplasty. The current article would focus on the various revascularization procedures.

The main goal of this article is to establish several new A-numerical radius equalities for n×ncirculant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices, where Ais the n×ndiagonal operator matrix whose diagonal entries are positive bounded operator A. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Further, some pinching type A-numerical radius inequalities for n×nblock operator matrices are given. Some equality conditions are also given. We also provide a concluding section, which may lead to several new problems in this area.