## Fixed point and interpolation point set of a positive linear operator on $$C(\overline{D})$$.(English)Zbl 1240.41069

The aim of this paper is to establish for a bounded open convex subset $$D\subset \mathbb{R}^p$$ and a positive linear operator $$A:\mathbb{C}(\overline{D})\to\mathbb{C}(\overline{D})$$ relations between the mixed-extremal point set of $$D$$, the fixed point set and the interpolation point set of $$A$$.
Let $$D$$ have nonempty interior, then a point $$x^0=(x_1^0,\dots,x_p^0)\in\partial D$$ is called mixed-extremal if, for each $$i\in\{1,\dots,p\}$$, $$x_i^0$$ is an extremal point of the ordered set $$(\{x_i\mid (x_1,\dots, x_p)\in D\},\leq_{\mathbb{R}})$$. The set of all mid-extremal points of $$D$$ is denoted by $$(ME)_D$$.
Moreover, a point $$x\in\overline{D}$$ is an interpolation point of $$A$$ if $$A(f)(x)=f(x)$$ for all $$f\in\mathbb{C}(\overline{D})$$; a subset $$E\subset\overline{D}$$ is an interpolation set of $$A$$ if $$A(f)|_E=f|_E$$.
The main results are as follows.
{(1)} Suppose that $$A$$ is an increasing operator and $$\Pi_1(\overline{D})\subset F_A$$, then $$(ME)_D$$ is an interpolation set of $$A$$.
($$\Pi_1$$ denotes the set of all polynomials of degree at most $$1$$; $$F_A$$ is not defined in the paper).
{(2)} If $$A$$ is an increasing operator and $$\Pi_0(\overline{D})\subset F_A$$, then $$E:=\{x\in (ME)_D\mid A(q_i)(x)=x_i\}$$ is an interpolation set of $$A$$.
{(3)} If $$A$$ is an increasing operator and $$q_1,\dots,q_p\in F_A$$, then $$E:=\{x\in (ME)_D\mid A(\tilde{1}(x)=1\}$$ is an interpolation set of $$A$$.
{(4)} Suppose $$A:C[a,b]\to C[a,b]$$ is an increasing linear operator and $$e_0,e_2\in F_A$$, then: if $$A(e_1)(a)=a$$, then $$a$$ is an interpolation point of $$A$$; if $$A(e_1)(b)=b$$, then $$b$$ is an interpolation point of $$A$$.

### MSC:

 41A36 Approximation by positive operators 41A05 Interpolation in approximation theory 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) 46B70 Interpolation between normed linear spaces 30E05 Moment problems and interpolation problems in the complex plane

### Keywords:

extremal points; interpolation points; increasing operators