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...

A bounded linear operator T on a complex Banach space X is called an operator of Saphar type if its kernel is contained in its generalized range ${\bigcap }_{n=1}^{\infty }{T}^n\left(X\right)$ and T is relatively regular. For T of Saphar type we determine the supremum of all positive numbers δ such that T - λI is of Saphar type for |λ| < δ.

Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\&lt;p\&lt;\infty$ and $v\ge 0$, we show there exists $C\&gt;0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\&gt;0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\&lt;\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

Let A be a pseudodifferential operator on ${ℝ}^N$ whose Weyl symbol a is a strictly positive smooth function on $W={ℝ}^N\times {ℝ}^N$ such that $|{\partial }^{\alpha }a|\le C_{\alpha }{a}^{1-\varrho }$ for some ϱ>0 and all |α|>0, ${\partial }^{\alpha }a$ is bounded for large |α|, and $li{m}_{w\to \infty }a\left(w\right)=\infty$. Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.

Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and $\left|\right|T^{n+1}-Tⁿ\left|\right|\asymp \left(n\ge 1\right)$. This answers Zemánek’s question on the time regularity property.

Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the...