A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal

preprint2020arXivOpen access

In this note, we give an example of a densely defined closed one-to-one paranormal operator $T$ whose adjoint is also injective and paranormal, but $T$ fails to be normal.

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A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal

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A closed densely defined operator TTT such that both TTT and T∗T^*T∗ are injective and paranormal yet TTT is not normal

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A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal