It makes sense to me that L-Theanine would help lessen positive symptoms of schizophrenia (SZ). Excessive dopaminergic activity (often re-uptake dysfunction at dopamine [DA] neurons) is often hypothesised (and sometimes accepted) as underpinning at least a component of what causes SZic positive symptoms. Certainly, excessive DA could theoretically (neuroscience) potentiate the onset of hallucinations.
Furthermore, excessive DA (particularly when chronic) can have an impact on the activity of other Monoamines, particularly the Catecholamines and in this case, most importantly Norepinephrine (NE). More DA will typically result in the production of more NE, which can increase adrenal/sympathetic tone, which describes the body's subconscious circuitry that tunes and balances the "fight or flight" and "rest and digest" conditions. These circuits are summarised in medical science as the Sympathetic Nervous System (SNS) and Parasympathetic Nervous System (PNS).
It stands to reason that a person with pervasive positive SZ symptoms has likely had chronically elevated DA levels and it is thereby rational to conclude that such a person may also be SNS dominant; meaning a chronic and constant state of excessive adrenal activity.
Assuming the person's adrenals were still functioning adequately, then it would also stand to reason that a nuero-inhibitory supplement such as L-Theanine would help reduce the impact of SNS induced anxiety, thus having a flow-on effect onto paranoia and finally, delusions.
L-Theanine is widely accepted (particularly anecdotally) to be a highly effacious adjunct complimenting stimulant medication treatment for ADHD diagnoses. In other words, nervous system stimulant drugs raise adrenal function (especially NE activity) and L-Theanine helps offset some of the jitters and anxious thinking that can result.