Positive Operators: Definition?

Question

Definitions

Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$

Consider selfadjoint operators $A=A^*\in\mathcal{A}$.

Define positive elements by: $$A\geq0:\iff\sigma(A)\geq0$$ and positive operators by: $$A\geq0:\iff\mathcal{W}(A)\geq0$$

Problem

Do the numerical range and spectrum coincide: $$A=A^*:\quad\langle\sigma(A)\rangle=\overline{\mathcal{W}(A)}$$

Attempt

For bounded operators one has at least: $$\|A\|<\infty:\quad\sigma(A)\subseteq\overline{\mathcal{W}(A)}$$ So any positive operator is a positive element; but what about the converse?

Answer 1

For unbounded normal operators one has: $$NN^*=N^*N:\quad\langle\sigma(N)\rangle=\overline{\mathcal{W}(N)}$$ (See summary on: Spectrum vs. Numerical Range)

So the notions of positivity agree!