In this article, we give a representation of bounded complex linear operators that preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is, moreover, positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group antihomomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realized by the idempotent preserving operators.

We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

Let $\alpha < \omega _1$ be a prime component, and let $X$ and $Y$ be metric spaces. In [8], it was shown that if $C_p(X)$ and $C_p(Y)$ are linearly homeomorphic, then the scattered heights $\kappa (X)$ and $\kappa (Y)$ of $X$ and $Y$ satisfy $\kappa (X) \leq \alpha $ if and only if $\kappa (Y) \leq \alpha $. We will prove that this also holds if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic and that these results do not hold for arbitrary Tychonov spaces. We will also prove that if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic, then $\kappa (X) < \alpha $ if and only if $\kappa (Y) < \alpha $, which was shown in [9] for $\alpha = \omega $. This last statement is not always true for linearly homeomorphic $C_p(X)$ and $C_p(Y)$. We will show that if $\alpha = \omega ^{\mu }$ where $\mu < \omega _1$ is a successor ordinal, it is true, but for all other prime components, this is not the case. Finally, we will prove that if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic, then $X$ is scattered if and only if $Y$ is scattered. This result does not directly follow from the above results. We will clarify why the results for linearly homeomorphic spaces $C_p^*(X)$ and $C_p^*(Y)$ do require a different and more complex approach than the one that was used for linearly homeomorphic spaces $C_p(X)$ and $C_p(Y)$.

In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.

In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$, the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$, satisfies the inequality $CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where $K(T)$ denotes the Kreiss constant of T and $C>0$ is an absolute constant. He also proved that for $r\ll 1/n,$ the latter bound is asymptotically sharp as $n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit $n\times n$ Toeplitz matrices with arbitrary singleton spectrum $\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to $\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to $\infty $.

Let f be an isolated singularity at the origin of $\mathbb {C}^n$. One of many invariants that can be associated with f is its Łojasiewicz exponent $\mathcal {L}_0 (f)$, which measures, to some extent, the topology of f. We give, for generic surface singularities f, an effective formula for $\mathcal {L}_0 (f)$ in terms of the Newton polyhedron of f. This is a realization of one of Arnold’s postulates.

In this article, we study the Bohr operator for the operator-valued subordination class $S(f)$ consisting of holomorphic functions subordinate to f in the unit disk $\mathbb {D}:=\{z \in \mathbb {C}: |z|<1\}$, where $f:\mathbb {D} \rightarrow \mathcal {B}(\mathcal {H})$ is holomorphic and $\mathcal {B}(\mathcal {H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal {H}$. We establish several subordination results, which can be viewed as the analogs of a couple of interesting subordination results from scalar-valued settings. We also obtain a von Neumann-type inequality for the class of analytic self-mappings of the unit disk $\mathbb {D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator-valued polyanalytic functions in certain proper simply connected domains in $\mathbb {C}$. We obtain Bohr radius for the operator-valued polyanalytic functions of the form $F(z)= \sum _{l=0}^{p-1} \overline {z}^l \, f_{l}(z) $, where $f_{0}$ is subordinate to an operator-valued convex biholomorphic function, and operator-valued starlike biholomorphic function in the unit disk $\mathbb {D}$.