Norm and Minimum Attainment of Foguel-Type Operators
Abstract
In this article, we study Foguel-type operators EuScriptF= V1* & T 0 & V2 on H1⊕ H2, where each Hj is a separable Hilbert space, Vj∈ ℬ(Hj) is an isometry for j=1,2, and T∈ ℬ(H2,H1). We examine how the norm and minimum attainment properties of EuScriptF depend on the corner entry T and on the defect spaces of the isometries. The norm formula |EuScriptF|=(|T|+√(|T|2+4))/2 holds for every T when V1,V2 are isometries, and more generally when V2 is a contraction that is isometric on the range of T*. Moreover, EuScriptF attains its norm if and only if T does. We characterize the positive 2× 2 block operator matrices that are absolutely norm attaining and absolutely minimum attaining, and use this to prove that EuScriptF is absolutely norm attaining if and only if it is absolutely minimum attaining if and only if dim N(V1*)<∞ and T has finite rank. The closure of the absolutely norm attaining class is described by dim N(V1*)<∞ and compactness of T. Further, we establish that the reduced minimum modulus is γ(EuScriptF)=(√(|T|2+4)−|T|)/2=1/|EuScriptF|, and EuScriptF attains it if and only if T is norm attaining. We also show that EuScriptF is Fredholm if and only if dim N(Vj*)<∞ for j=1,2, with index dim N(V1*)−dim N(V2*), and we give its Moore--Penrose inverse. Finally, for V1=V2=Tz and more generally Vj=Tθj with θj inner, and Toeplitz or Hankel corner entries, the results become explicit conditions on the symbol.
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BibTeX
@misc{nag2026norm,
title = {{Norm and Minimum Attainment of Foguel-Type Operators}},
author = {Puspendu Nag and Ramesh Golla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42102},
eprint = {2610.42102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42102},
}
RIS
TY - UNPB
TI - Norm and Minimum Attainment of Foguel-Type Operators
AU - Puspendu Nag
AU - Ramesh Golla
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42102
UR - https://arxiv.org/abs/2610.42102
AB - In this article, we study Foguel-type operators EuScriptF= V_1^* & T 0 & V_2 on H_1⊕ H_2, where each H_j is a separable Hilbert space, V_j∈ ℬ(H_j) is an isometry for j=1,2, and T∈ ℬ(H_2,H_1). We examine how the norm and minimum attainment properties of EuScriptF depend on the corner entry T and on the defect spaces of the isometries. The norm formula |EuScriptF|=(|T|+√(|T|^2+4))/2 holds for every T when V_1,V_2 are isometries, and more generally when V_2 is a contraction that is isometric on the range of T^*. Moreover, EuScriptF attains its norm if and only if T does. We characterize the positive 2× 2 block operator matrices that are absolutely norm attaining and absolutely minimum attaining, and use this to prove that EuScriptF is absolutely norm attaining if and only if it is absolutely minimum attaining if and only if dim N(V_1^*)<∞ and T has finite rank. The closure of the absolutely norm attaining class is described by dim N(V_1^*)<∞ and compactness of T. Further, we establish that the reduced minimum modulus is γ(EuScriptF)=(√(|T|^2+4)−|T|)/2=1/|EuScriptF|, and EuScriptF attains it if and only if T is norm attaining. We also show that EuScriptF is Fredholm if and only if dim N(V_j^*)<∞ for j=1,2, with index dim N(V_1^*)−dim N(V_2^*), and we give its Moore--Penrose inverse. Finally, for V_1=V_2=T_z and more generally V_j=T_θ_j with θ_j inner, and Toeplitz or Hankel corner entries, the results become explicit conditions on the symbol.
ER -