Linear maps preserving essentially numerical range and essentially unitary operator (Q326881)

Summary: Let \(\mathcal{H}\) be an infinite dimensional separable complex Hilbert space and \(\mathcal{B}(\mathcal{H})\) the algebra of all bounded linear operators on \(\mathcal{H}\). We give the concrete forms of maps \(\phi: \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H})\) that preserve the essential numerical range, the essential numerical radius in both direction or essential unitary operator in one directions.