Is there a gizmo which can connect to my living room wall TV aerial
socket which will send the TV signal to my bedroom TV"s which
presently have set top aerials on them. At the same time I still want to
maintain the lead from the walll socket to my living room TV?

Well, yeah, sort-of, but it's a bad idea.

In order to do this, you'd need a broadband amplifier, and an indoor
antenna which would retransmit the whole TV spectrum.

The problem with this approach is that if you amplify the signal from
the aerial enough to reach the bedroom TVs with a useful strength,
there's a *very* good chance that it would be strong enough to reach
back up to the aeriel and be picked up again (mixing with the incoming
signal from the TV transmitters). At best, this would cause image
degradation and "ghosts" (if analog) or a higher bit-error rate with
video dropouts (if digital). If worse, it would create a horrendous
RF feedback problem which would obliterate television reception
throughout your house, and possibly affect nearby houses as well.

It's sort of the radio-frequency equivalent of using a PA system, with
the microphone too close to one of the speakers... it lets out a
terrible "howl" of feedback. This is not what you want.

Also, it's probably illegal to do this in your jurisdiction, since you
would in effect be creating an unlicensed over-the-air television
transmission system.

You're much better off using a "wired" signal distribution system...
typically, with 75-ohm coaxial cable... and running a coax to each
individual television set. You would need a signal splitter (passive),
or a powered "distribution amplifier" if your aerial signal isn't
strong enough.

You can (in almost all cases) disconnect the set-top aerial, and
connect the coax to the TV instead (some have 75-ohm coax inputs, and
others have a pair of screw terminals for a 300-ohm connection - you
would need a 75-to-300-ohm "balun" to make the connection).

--
Dave Platt AE6EO
Friends of Jade Warrior home page: http://www.radagast.org/jade-warrior
I do _not_ wish to receive unsolicited commercial email, and I will
boycott any company which has the gall to send me such ads!