Let ℬ(ℋ) be the algebra of all bounded linear operators acting on a complex Hilbert space ℋ , and denote by w(T) the numerical radius of any operator T∈ℬ(ℋ) . When ℋ is an infinite-dimensional separable space, we described, in [Linear Algebra Appl., 714 (2025), 15-27], the form of all bijective linear maps Φ on ℬ(ℋ) for which w(Φ (T))=w(Φ (S)) whenever T, S∈ℬ(ℋ) are two unitarily similar operators. In this note, we provide a short, simple proof of such a result, and discuss an analog result when ℋ is a finite-dimensional Hilbert space.
We investigate singularities of height functions on generalized cuspidal edges. In particular, we study singularities of height functions in the normal and conormal directions. Moreover, we examine the bifurcation set of the family of height functions.
A Lehmer number modulo an odd prime number p is a residue class a ∈𝔽_p^× whose multiplicative inverse a̅ is of opposite parity with a. Lehmer numbers that are also quadratic residues (respectively, non-residues, primitive roots and non-primitive roots) are called Lehmer quadratic residues (respectively, Lehmer non-residues, Lehmer primitive roots and Lehmer non-primitive roots). In this article, we prove that for any prime p > e^27 (respectively, for any prime p satisfying loglog p > log 2.62/(1/2 - ϵ ) with ϵ∈ [1/5, 1/2) and ϕ (p-1)/ (p-1) ≤ 1/2 - ϵ ), there exists Lehmer quadratic residue and Lehmer non-residue (respectively, Lehmer non-residue which are non-primitive root) modulo p. We also provide explicit estimates for the difference between the number of such numbers and their “expected number”. We also study the Lehmer number problem by varying primes p. In particular, for a fixed integer a ≥ 2 , we prove the existence of infinitely many prime numbers p for which a is a Lehmer number modulo p. In particular, we obtain a bound for the relative density of the set of all primes for which a is a Lehmer number, in terms of ϕ (a) . Moreover, when a = 2 or 3, we explicitly find congruence classes for such primes.
Let ℋ and 𝒦 be complex separable Hilbert spaces, and consider T= [ aI_ℋ A; cA^* bI_𝒦 ], A∈ℬ(𝒦,ℋ), a,b,c∈ℂ. Using the polar decomposition of A, we separate the coupled part of T from the scalar components supported on the defect spaces A^* and A . The spectral theorem then reduces the coupled part to the family M_t= [ a t; ct b ], t∈σ( . |A|| _( A)^⊥) . This yields exact formulas for both the minimum modulus and the essential minimum modulus of T. We characterize minimum attainment in terms of the spectral measure of the reduced modulus and obtain necessary and sufficient conditions for boundedness below and upper semi-Fredholmness. We further show that a strict gap between the two moduli forces the ordinary minimum to be an isolated eigenvalue of |T| of finite multiplicity. Applications are given to triangular operators, partial isometries, compact off-diagonal operators, and Buckholtz-type operators.
