It was thought (by many) to have been written by Michael.

The sentence is grammatically correct, Jackson, and completely in accordance with English grammatical rules and practices. To have been written is a passive perfect infinitive. There are only two infinitives in English and they can be used in both active and passive. As a rule, the present infintive refers to the present and the future and the perfect infinitive refers to the past:

Active, present infinitive:
I would write it down.
He may come tomorrow.
I want to say something.
It's easy to speak English.

Passive, present infinitive:
It should be written down.
He won't be caught.
To be seen is not the same as to be caught.

Active, perfect infintive:
I would have written it down.
To have loved and lost is better than never to have loved at all.

Passive, perfect infinitive:
It was thought to have been written by Michael.
Something should have been done earlier.
They may have been seen at the station.

CB

Jackson6612 wrote:

Is what I say above correct? Is my description correct?

I see nothing worth arguing.

CB

What kind of mood does ''should'' represent?

CB wrote: ''As a rule, the present infintive refers to the present and the future and the perfect infinitive refers to the past.''

I'm changing my original sentence to the present tense:

It is thought (by many) to have been written by Michael.

''to have been'' is perfect infinitive but here it refers to the present. Isn't it in conflict with what CB said above?