The existence of solutions is studied for certain nonlinear differential equations with both linear and nonlinear conditions

Let T₁,...,Tₙ be bounded linear operators on a complex Hilbert space H. Then there are compact operators K₁,...,Kₙ ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T₁-K₁,...,Tₙ-Kₙ) equals the joint essential numerical range of (T₁,...,Tₙ). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ ℕ then there exists a compact operator K ∈ B(H) such that $\left|\right|(S-K)ⁿ\left|\right|={\left|\right|Sⁿ\left|\right|}_e$. This generalizes results of C. L. Olsen.

Let W(A) and $W_e\left(A\right)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_e\left(A\right)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_e\left(A\right)$ can be obtained as the intersection of all sets of the form $cl\left(W(A₁,...,A_{i+1},A_i+F,A_{i+1},...,Aₘ)\right)$, where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in $W_e\left(A\right)$ as star centers....

We show that every Jordan isomorphism between CSL algebras is the sum of an isomorphism and an anti-isomorphism. Also we show that each Jordan derivation of a CSL algebra is a derivation.

The main result of the paper characterizes continuous local derivations on a class of commutative Banach algebra $A$ that all of its squares are positive and satisfying the following property: Every continuous bilinear map $\Phi$ from $A\times A$ into an arbitrary Banach space $B$ such that $\Phi (a,b)=0$ whenever $ab=0$, satisfies the condition $\Phi (ab,c)=\Phi (a,bc)$ for all $a,b,c\in A$.

Let $T\in ℒ\left(X\right)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f\left(z\right)\mapsto (z-T)f\left(z\right)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and...

Let $𝒜$ be a commutative complex semisimple Banach algebra. Denote by $\mathrm{kh}\left(\mathrm{soc}\right(𝒜\left)\right)$ the kernel of the hull of the socle of $𝒜$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $𝒜$. This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.