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We say that an operator T∈B(H)T \\in B(\\mathcal {H}) is complex symmetric if there exists a conjugate-linear, isometric involution C:H→HC:\\mathcal {H}\\rightarrow \\mathcal {H} so that T=CT∗CT = CT^*C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim⁡ker⁡T,dim⁡ker⁡T∗)(\\dim \\ker T, \\dim \\ker T^*).

A bounded linear operator TT on a complex Hilbert space H\\mathcal {H} is called complex symmetric if T=CT∗CT = CT^*C, where CC is a conjugation (an isometric, antilinear involution of H\\mathcal {H}). We prove that T=CJ|T|T = CJ|T|, where JJ is an auxiliary conjugation commuting with |T|=T∗T|T| = \\sqrt {T^*T}. We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T=CJ|T|T = CJ|T| also extends to the class of unbounded CC-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in Chávez, Garcia, and Hurley (2023, Canadian Mathematical Bulletin 66, 808–826) from exponent $d\\geq 2$ to $d \\geq 1$ . Our proofs are much simpler than the originals: they do not require Lewis’ framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.

We introduce a family of norms on the $n \\times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

There have been many articles in the MONTHLY on quotient sets over the years. We take a first step here into the p-adic setting, which we hope will spur further research. We show that the set of quotients of nonzero Fibonacci numbers is dense in the p-adic numbers for every prime p.

We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.

We develop a method for calculating the norm and the spectrum of the modulus of a Foguel operator. In many cases, the norm can be computed exactly. In others, sharp upper bounds are obtained. In particular, we observe several connections between Foguel operators and the Golden Ratio.

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.