Causality, Linearity, and Time Invariance for Systems Described by Linear Constant Coefficient Differential Equations
Note that the system is not uniquely characterized by the linear differential operator. You need auxiliary conditions, and dependent on these conditions, the system may be linear, causal and time-invariant or not. So if you just use the operator $L$ to show that the system is linear then you're actually not showing anything because $L$ is a linear operator by definition.

The output of a system described by a linear constant coefficient differential equation can be split up into two contributions: the zero-state response (ZSR) and the zero-input response (ZIR). The ZSR is the response of the system with zero initial conditions, and, consequently, the ZSR is fully determined by the input signal. The ZIR response is the response of the system caused by non-zero initial conditions, and it is independent of the input signal. Since non-zero initial conditions cause an output term that does not depend on the input signal, scaling or shifting the input signal will not affect that term. Consequently, a system with non-zero initial conditions can neither be linear nor time-invariant, because if it were, the output should scale and shift according to the scaling and shifting of the input signal.

Causality simply means that the system's response at a given time does not depend on the future of the input signal. Clearly, causality is a property of the system, so it cannot depend on the input signal. A causal linear time-invariant (LTI) system is characterized by an impulse response $h(t)$ that satisfies $h(t)=0$ for $t&lt;0$, and it is irrelevant whether the input signal is zero before a certain time or not.

Also take a look at this related answer.