Hi Fedde
The average PJ already operates in a "pressurised environment" , you have to think in absolute terms .
A PJ designed for operation at ~34,000 ft ( atmospheric pressure ~3.6 psia ) , when operated at sea level at 14.7 psia would be in a 4:1 pressure ratio environment .
As for having a gas turbine supplying the "compressed air" , its possible , but the PJ would only be replacing the normal combustor in supplying heat input .
You'd probably need to have the gas turbine running first , then slowly reduce the normal fuel flow as the fuel to the PJ was introduced so as to maintain maximum turbine inlet temperatures ( TIT) .
Trying to change over from normal flametube combustion to PJ combustion would be a nightmare , any drop in TIT would see a reduction in pressure ratio/rpm and mass flow , and conversely any increase in TIT an increase in pressure ratio/rpm and mass flow if the the turbine stage was able to swallow the extra flow .
Using a valveless PJ would only be complicating matters I feel , thats why I used a valved engine to run my Turbo Prop PJ experiment .
If the running PJ contributed some extra "total pressure" downstream of itself and before the turbine stage , this would cause rotor acceleration , so trying to keep rpm at below maximum rpm ( N1 100%) whilst still trying to maintain a high pressure ratio from the compressor would be rather difficult.
Summary , running a PJ in a pressurised environment should be possible , having a gas turbine supplying that environment ,.... problematic , and one has to ask ...... for what ?? maybe a slight increase in potential thrust output from the gas turbine engine , but with a huge amount of complication, internal frictional air/gas losses, increased weight and frontal "envelope" .................not worth all the trouble

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PJs are PJs , gas turbines are gas turbines , combining the two .....................interesting in theory .............but in the real world , not practical ...............just my personal opinion .................LOL.....I'd be happy to see someone shoot it to pieces though , I tried... I failed

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Cheers
John
Cheers
John