Rigidity of the structured singular value and applications
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2026
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| author | Pal, Sourav Tomar, Nitin |
| author_facet | Pal, Sourav Tomar, Nitin |
| contents | The structured singular value $μ_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by \[ μ_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})), \] and $μ_E(A)=0$ if there is no $X \in E$ with $\det(I_n-AX)=0$. It is well-known that $μ_E(A)$ coincides with the spectral radius $r(A)$ when $E=\{cI_n: c \in \mathbb C \}$ and $μ_E(A)=\|A\|$ when $E=M_n(\mathbb C)$, for all $A\in M_n(\mathbb C)$. Also, for any linear subspace $E$ satisfying $\{cI_n: c \in \mathbb C \} \subseteq E \subseteq M_n(\mathbb C)$, we have $r(A)\leq μ_E(A) \leq \|A\|$. We prove that if $E=\{cI_n: c \in \mathbb C \}$ and $F$ is any linear subspace of $M_n(\mathbb C)$ containing $E$, then $μ_E=μ_F$ if and only if $E=F$. We prove the exact same rigidity theorem for the linear subspace consisting of the diagonal matrices of order $n$. On the contrary, when $E=M_n(\mathbb C)$, we show that there is a proper subspace $F$ of $M_n(\mathbb C)$, viz. the space of symmetric matrices such that $μ_E=μ_F=$ operator norm. Further, we characterize all linear subspaces $F\subseteq M_n(\mathbb C)$ such that $μ_F$ coincides with the operator norm. Next, we show that in general there is no subspace $E$ of $M_n(\mathbb C)$ such that $μ_E=$ the numerical radius, not even for $M_2(\mathbb C)$. We establish the rigidity of the structured singular value for each of the subspaces $E$ of $M_2(\mathbb C)$ such that the corresponding $μ_E$-unit ball induces the domains -- symmetrized bidisc, tetrablock, pentablock, hexablock. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26312 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity of the structured singular value and applications Pal, Sourav Tomar, Nitin Functional Analysis Complex Variables Operator Algebras The structured singular value $μ_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by \[ μ_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})), \] and $μ_E(A)=0$ if there is no $X \in E$ with $\det(I_n-AX)=0$. It is well-known that $μ_E(A)$ coincides with the spectral radius $r(A)$ when $E=\{cI_n: c \in \mathbb C \}$ and $μ_E(A)=\|A\|$ when $E=M_n(\mathbb C)$, for all $A\in M_n(\mathbb C)$. Also, for any linear subspace $E$ satisfying $\{cI_n: c \in \mathbb C \} \subseteq E \subseteq M_n(\mathbb C)$, we have $r(A)\leq μ_E(A) \leq \|A\|$. We prove that if $E=\{cI_n: c \in \mathbb C \}$ and $F$ is any linear subspace of $M_n(\mathbb C)$ containing $E$, then $μ_E=μ_F$ if and only if $E=F$. We prove the exact same rigidity theorem for the linear subspace consisting of the diagonal matrices of order $n$. On the contrary, when $E=M_n(\mathbb C)$, we show that there is a proper subspace $F$ of $M_n(\mathbb C)$, viz. the space of symmetric matrices such that $μ_E=μ_F=$ operator norm. Further, we characterize all linear subspaces $F\subseteq M_n(\mathbb C)$ such that $μ_F$ coincides with the operator norm. Next, we show that in general there is no subspace $E$ of $M_n(\mathbb C)$ such that $μ_E=$ the numerical radius, not even for $M_2(\mathbb C)$. We establish the rigidity of the structured singular value for each of the subspaces $E$ of $M_2(\mathbb C)$ such that the corresponding $μ_E$-unit ball induces the domains -- symmetrized bidisc, tetrablock, pentablock, hexablock. |
| title | Rigidity of the structured singular value and applications |
| topic | Functional Analysis Complex Variables Operator Algebras |
| url | https://arxiv.org/abs/2603.26312 |