REPRESENTATION OF COMPACT OPERATORS BETWEEN BANACH SPACES G. Ramesh, M. Veena Sangeetha, Shanola S. Sequeira Journal of Operator Theory, 2025 In this article, we give a representation for compact operators acting between reflexive Banach spaces, which generalizes the representation given by Edmunds et al. for compact operators between reflexive Banach spaces with strictly convex duals. Further, we give a representation for a class of operators on Banach spaces, that is comparable to the classical spectral representation for compact normal operators on Hilbert spaces. Finally, we give an example to illustrate our main result.
On k-strong convexity in banach spaces Journal of Convex Analysis, 2021
Geometric and fixed point properties in products of normed spaces M. VEENA SANGEETHA Bulletin of the Australian Mathematical Society, 2019 Given two (real) normed (linear) spaces $X$ and $Y$, let $X\\otimes _{1}Y=(X\\otimes Y,\\Vert \\cdot \\Vert )$, where $\\Vert (x,y)\\Vert =\\Vert x\\Vert +\\Vert y\\Vert$. It is known that $X\\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\\Vert \\cdot \\Vert _{E}$ on $\\mathbb{R}^{2}$, we let $X\\otimes _{E}Y=(X\\otimes Y,\\Vert \\cdot \\Vert )$ where $\\Vert (x,y)\\Vert =\\Vert (\\Vert x\\Vert _{X},\\Vert y\\Vert _{Y})\\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\\Vert \\cdot \\Vert _{E}$ is strictly monotone, then $X\\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.
On relative k-uniform rotundity, normal structure and fixed point property for nonexpansive maps Journal of Nonlinear and Convex Analysis, 2019
Uniform rotundity with respect to finite-dimensional subspaces Journal of Convex Analysis, 2018
Representation of Compact Operators between Banach spaces G Ramesh, MV Sangeetha, SS Sequeira arXiv preprint arXiv:2308.07756 , 2023 2023 Citations: 1
Geometry of product spaces MV Sangeetha Journal of Mathematical Analysis and Applications 503 (1), 125285 , 2021 2021 Citations: 3
On k-strong convexity in Banach spaces MV Sangeetha, M Radhakrishnan, S Kar J. Convex Anal. 28 (4), 1193-1210 , 2021 2021 Citations: 4
Geometric and fixed point properties in products of normed spaces MV Sangeetha Bulletin of the Australian Mathematical Society 99 (2), 262-273 , 2019 2019 Citations: 4
On relative k-uniform rotundity, normal structure and fixed point property for nonexpansive maps MV Sangeetha JOURNAL OF NONLINEAR AND CONVEX ANALYSIS 20 (2), 321-336 , 2019 2019 Citations: 3
Uniform rotundity with respect to finite-dimensional subspaces MV Sangeetha, P Veeramani J. Convex Anal 25 (4), 1223-1252 , 2018 2018 Citations: 13
Normal structure and invariance of Chebyshev center under isometries MV Sangeetha, P Veeramani Journal of Mathematical Analysis and Applications 436 (1), 611-619 , 2016 2016 Citations: 3
MOST CITED SCHOLAR PUBLICATIONS
Uniform rotundity with respect to finite-dimensional subspaces MV Sangeetha, P Veeramani J. Convex Anal 25 (4), 1223-1252 , 2018 2018 Citations: 13
On k-strong convexity in Banach spaces MV Sangeetha, M Radhakrishnan, S Kar J. Convex Anal. 28 (4), 1193-1210 , 2021 2021 Citations: 4
Geometric and fixed point properties in products of normed spaces MV Sangeetha Bulletin of the Australian Mathematical Society 99 (2), 262-273 , 2019 2019 Citations: 4
Geometry of product spaces MV Sangeetha Journal of Mathematical Analysis and Applications 503 (1), 125285 , 2021 2021 Citations: 3
On relative k-uniform rotundity, normal structure and fixed point property for nonexpansive maps MV Sangeetha JOURNAL OF NONLINEAR AND CONVEX ANALYSIS 20 (2), 321-336 , 2019 2019 Citations: 3
Normal structure and invariance of Chebyshev center under isometries MV Sangeetha, P Veeramani Journal of Mathematical Analysis and Applications 436 (1), 611-619 , 2016 2016 Citations: 3
