Narrow Results

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Gait assessment is important for identification of potential faller among the elderly populations. Slope walking is associated with fall risk factor and elderly women have higher fall rate compared with elderly men. Therefore, this study investigated gait characteristics of elderly women in overground and slope walkway conditions. Thirty healthy elderly women (15 younger-elderly women and 15 older-elderly women) walked along the linear walkway including three walking conditions (overground, ascent and descent conditions). Temporal gait variables and normalized peak vertical GRF (ground reaction force) variables were derived from commercial motion analysis software. Repeated-measures analysis of variance (ANOVA) was evaluated to compare mean differences of the three conditions and mean difference between younger and older elderly women. All gait characteristics were significantly different from the slope walking conditions ($p<0.05$). Elderly women walked with longer loading response and mid stance phase during descent walking. Also, ascent walking induced a longer terminal stance phase. Interactions of age and walkway conditions were also significant in vertical GRF, where older-elderly women were greater than younger-elderly women in ascent walkway condition ($p<0.01$) and in descent walkway condition ($p=0.05$). These findings suggest that specific-walkway condition should be considered for fall prevention and clinical interventions in elderly women.

Let $\mathcal{ℬ}(X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space $X$. We prove that a bijective bicontinuous map $\mathrm{\Phi }$ on $\mathcal{ℬ}(X)$ preserves the difference of group invertible operators in both directions if and only if $\mathrm{\Phi }$ is either of the form $\mathrm{\Phi }(T)=\alpha AT{A}^{-1}+\mathrm{\Phi }(0)$ or of the form $\mathrm{\Phi }(T)=\alpha B{T}^{\ast }{B}^{-1}+\mathrm{\Phi }(0)$, where $\alpha$ is a nonzero scalar, $A:X\to X$ and $B:X^{\ast }\to X$ are two bounded invertible linear or conjugate linear operators.

A bounded operator $T$ in a Banach space $X$ is said to satisfy the descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the descent spectrum of $T$ as an element of the algebra $B(X)$ of all bounded linear operators on $X$. In this paper, we give some conditions under which the equality ${\sigma }_{\mathrm{\text{desc}}}(T)={\sigma }_{\mathrm{\text{desc}}}(T,B(X))$ holds for a single operator $T$.

The purpose of this paper is to present new additive results for $n$-quasi-isometries operators on Hilbert spaces. In particular Precisely, we focus on the study of the descent of a $n$-quasi-isometry operator. Some spectral properties for this class of operators and decomposition theorems are also given. Part of the results proved in this paper improve and generalize some results known for isometries and quasi-isometries.