LTI system $y(t)=x(t-T)$ with or without memory
From Signals and Systems/Time Domain Analysis -- Memory


  A system is said to have memory if the output from the system is
  dependent on past inputs (or future inputs) to the system. A system is
  called memoryless if the output is only dependent on the current
  input.


According to some textbooks, the notion of memory is stricter, and requires causality of the system. To that respect:


if $T\ge 0$, the system is causal. If $T=0$, it only use the current time (memoryless).   If $T&gt;0$, the output need a memory of the past of input $x$
if $T&lt; 0$, the system is non-causal.  The output needed a memory of the future of input $x$. You can either say "non applicable", or still term it "memory".


References (especially the first, with many practical exemples):


Lecture III: Systems and their properties by Maxim Raginsky
$y(n)=x(-n)$, causal or not, memory or memoryless?