We show that every spectrally bounded linear map Φ from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if Φ₂ is spectrally bounded, then Φ is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection Φ from ℬ(H) onto ℬ(K), where H and K are infinite-dimensional...

The characterization of the domain of the Friedrichs extension as a restriction of the maximal domain is well known. It depends on principal solutions. Here we establish a characterization as an extension of the minimal domain. Our proof is different and closer in spirit to the Friedrichs construction. It starts with the assumption that the minimal operator is bounded below and does not directly use oscillation theory.

In this paper we consider a class of three-term recurrence relations, whose associated tridiagonal matrices are subnormal operators. In this cases, there are measures associated to the polynomials given by such relations. We study the support of these measures.

Let $A$ and $B$ be two Archimedean vector lattices and let ${\left(A^{\text{'}}\right)}_n^{\text{'}}$ and ${\left(B^{\text{'}}\right)}_n^{\text{'}}$ be their order continuous order biduals. If $\Psi :A\times A\to B$ is a positive orthosymmetric bimorphism, then the triadjoint ${\Psi }^{***}:{\left(A^{\text{'}}\right)}_n^{\text{'}}\times {\left(A^{\text{'}}\right)}_n^{\text{'}}\to {\left(B^{\text{'}}\right)}_n^{\text{'}}$ of $\Psi$ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.

Let ${\varphi }_1,\cdots ,{\varphi }_n$ be real homogeneous functions in $C^{\infty }({ℝ}^n-\left\{0\right\})$ of degree $k\ge 2$, let $\varphi \left(x\right)=({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right))$ and let $\mu$ be the Borel measure on ${ℝ}^{2n}$ given by $$\mu \left(E\right)={\int }_{{ℝ}^n}{\chi }_E(x,\varphi \left(x\right)){\left|x\right|}^{\gamma -n}\mathrm{d}x$$ where $\mathrm{d}x$ denotes the Lebesgue measure on ${ℝ}^n$ and $\gamma >0$. Let $T_{\mu }$ be the convolution operator $T_{\mu }f\left(x\right)=(\mu *f)\left(x\right)$ and let $$E_{\mu }=\{(1/p,1/q)\parallel T_{\mu }{\parallel }_{p,q}<\infty ,1\le p,q\le \infty \}.$$ Assume that, for $x\ne 0$, the following two conditions hold: $det\left({\mathrm{d}}^2\varphi \left(x\right)h\right)$ vanishes only at $h=0$ and $det\left(\mathrm{d}\varphi \right(x\left)\right)\ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_{\mu }$ is the empty set and if $\gamma \le n(k+1)/3$ then $E_{\mu }$ is the closed segment with endpoints $D=\left(1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\right)$ and $D^{\text{'}}=\left(\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\right)$. Also, we give some examples.

We characterize Banach lattices on which every positive almost Dunford-Pettis operator is weakly compact.